A Classification of Curtis-Tits Amalgams

Abstract

A celebrated theorem of Curtis and Tits on groups with finite BN-pair shows that these groups are determined by the local structure arising from their fundamental subgroups of ranks $1$ and $2$. This result was later extended to Kac-Moody groups by P. Abramenko and B. Mühlherr and Caprace. Their theorem states that a Kac-Moody group $G$ is the universal completion of an amalgam of rank two (Levi) subgroups, as they are arranged inside $G$ itself. Taking this result as a starting point, we define a Curtis-Tits structure over a given diagram to be an amalgam of groups such that the sub-amalgam corresponding to a two-vertex sub-diagram is the Curtis-Tits amalgam of some rank-$2$ group of Lie type. There is no a priori reference to an ambient group, nor to the existence of an associated (twin-) building. Indeed, there is no a priori guarantee that the amalgam will not collapse. We then classify these amalgams up to isomorphism. In the present paper we consider triangle-free simply-laced diagrams. Instead of using Goldschmidt’s lemma, we introduce a new approach by applying Bass and Serre’s theory of graphs of groups, not to the amalgams themselves but to a graph of groups consisting of certain automorphism groups. The classification reveals a natural division into two main types: “orientable” and “non-orientable” Curtis-Tits structures. Our classification of orientable Curtis-Tits structures naturally fits with the classification of all locally split Kac-Moody groups over fields with at least four elements using Moufang foundations. In particular, our classification yields a simple criterion for recognizing when Curtis-Tits structures give rise to Kac-Moody groups. The class of non-orientable Curtis-Tits structures is in some sense much larger. Many of these amalgams turn out to have non-trivial interesting completions inviting further study.

Subject Classification: [2010] Primary 20G35; Secondary 51E24, 20E42The authors wish to thank the Banff International Research Station, Banff, Canada for a very productive visit as part of their Research In Teams program (08rit130).

1 Introduction

Kac-Moody Lie algebras are infinite dimensional Lie algebras defined by relations analogous to the Serre relations for finite dimensional semi-simple Lie algebras. They have been introduced in the mid 1960s by V. Kac and R. Moody. The affine Kac-Moody and generalized Kac-Moody Lie algebras have extensive applications to theoretical physics, especially conformal field theory, monstrous moonshine and more.

Finite dimensional semi-simple Lie algebras admit Chevalley bases which allow the construction of Chevalley groups, Lie-type groups over arbitrary fields. By analogy, J. Tits gave a new interpretation of Kac-Moody groups as groups having a twin-root datum, which implies that they are symmetry groups of Moufang twin-buildings (see [27, 28]). In the case that the corresponding diagram is spherical, the corresponding group is a Chevalley group. These and other similar groups play a very important role in various aspects of geometric group theory. In particular, they provide examples of infinite simple groups (see for example [10, 11, 13, 16]).

A celebrated theorem of Curtis and Tits [14, 26] (later extended by Timmesfeld (see [22–25] for spherical groups), by Abramenko and Mühlherr [1] and Caprace [12] to two-spherical Kac-Moody groups) on groups with finite BN-pair states that the Kac-Moody groups are the universal completion of the concrete amalgam of the Levi components of the parabolic subgroups with respect to a given (twin-) BN-pair.

In case that the amalgam is unique, this suffices to recognize the group. In general however, this is an inconvenience since it is usually easy to recognize isomorphism classes of subgroups but perhaps not so easy to globally manage their embedding. This is the reason that one often restricts to the so called “split” Kac-Moody groups. However “twisted” versions of Kac-Moody groups do exist, as constructed in [16, 19] and they in turn give Curtis-Tits amalgams.

A natural question is therefore the following: how can one recognize these amalgams as abstract group amalgams? More generally one would like to classify all amalgams that are “locally” isomorphic to the usual Curtis-Tits ones and identify their universal completions. In this paper we use a variation of Bass-Serre theory to classify all Curtis-Tits structures over a field with at least four elements and with connected simply laced triangle-free Dynkin diagrams. As a by-product we obtain a description of all Kac-Moody groups in this case.

Throughout the paper $k$ will be a commutative field of order at least four. We need the restriction on the order for the classification of the amalgams. Precise definitions will be given in Sect. 3. Let $\varGamma$ be a Dynkin diagram over a set $I$. A Curtis-Tits (CT) structure over $k$ with Dynkin diagram $\varGamma$ is an amalgam ${\fancyscript{G}}=\{G_i,G_{i,j}\mid i,j\in I\}$ such that, for every $i,j\in I$, $\{G_i,G_j,G_{i,j}\}$ is a Curtis-Tits amalgam of type $\varGamma _{i,j}$. In case $\varGamma$ is simply-laced, this means that the rank-$1$ groups $G_i$ are isomorphic to $\mathrm{SL}_2(k)$, where $G_{i,j}=\langle G_i,G_j\rangle$, that $G_i$ and $G_j$ commute if $\{i,j\}$ is a non-edge in $\varGamma$ and their images in $G_{i,j}\cong \mathrm{SL}_3(k)$ form a standard pair (see Definition 3) if $\{i,j\}$ is an edge in $\varGamma$. Note that we do not impose any further conditions on the embedding maps $G_i\hookrightarrow G_{i,j}$.

We are only interested in CT structures that admit a non-trivial completion. The universal completion of a (non-collapsing) Curtis-Tits structure is called a Curtis-Tits group.

In fact, a slight extension of our methods allows to classify Curtis-Tits structures for a larger class of diagrams, including for instance all three-spherical Dynkin diagrams and, more general amalgams such as Phan type amalgams. However, in order to present these new methods and new results in a transparent manner, we chose to restrict to all simply-laced diagrams without triangles, just as Tits did in his classification of Moufang foundations for these diagrams in [28].

Curtis-Tits groups that are not Kac-Moody groups do exist. As an application in [5] we give constructions of all possible Curtis-Tits structures with diagram $\widetilde{A}_n$, realizing them as concrete amalgams inside their respective non-trivial completions, and giving sufficient conditions for these completions to be universal. In addition, we furnish geometric objects on which these groups act naturally. This leads us to describe two very interesting collections of groups. The first is a collection of twisted versions of the Kac-Moody group $\mathrm{SL}_n(k[t,t^{-1}])$ whose natural quotients are labeled by the cyclic algebras of center $k$. The corresponding twin-building is related to Drinfeld’s vector bundles over a non-commutative projective line. The second is a collection of Curtis-Tits groups that are not Kac-Moody groups. One of these maps surjectively to $\mathrm{Sp}_{2n}(q)$, $\varOmega _{2n}^{+}(q)$, and $\mathrm{SU}_{2n}(q^l)$, for all $l\ge 1$, making this family of unitary groups into a family of expanders [8].

The main result of the present paper is the following.

Theorem 1.1

Let $\varGamma$ be a connected simply laced Dynkin diagram with no triangles and $k$ a field with at least four elements. There is a natural bijection between isomorphism classes of CT-structures over the field $k$ on a graph $\varGamma$ of a given type and group homomorphisms $\varPhi :\pi (\varGamma , i_0)\rightarrow {\mathbb Z}_2\times \mathrm{Aut}(k)$.

Here, $\pi (\varGamma ,i_0)$ denotes the fundamental group of the graph $\varGamma$ with base point $i_0$. As mentioned above, the motivation for the work came from the Curtis-Tits amalgam presentations for Kac-Moody groups. In fact in the spherical case these were proved to be the only such amalgams. In general the proportion of orientable amalgams among all Curtis-Tits amalgams decreases with the rank of $\pi (\varGamma ,i_0)$. More precisely they are those amalgams in the theorem corresponding to maps $\varPhi$ so that $\mathrm{Im}(\varPhi ) \le \mathrm{Aut}(k)$. We call such amalgams orientable.

Corollary 1.2

Let $\varGamma$ be a connected simply laced Dynkin diagram with no triangles and $k$ a field with at least four elements. The universal completion of a Curtis-Tits structure over a commutative field $k$ and diagram $\varGamma$ is a central extension of a locally split Kac-Moody group over $k$ with Dynkin diagram $\varGamma$ (and ${\fancyscript{G}}$ is the Curtis-Tits amalgam for this group) if and only if ${\fancyscript{G}}$ is orientable.

Note that for example in [1, 12, 28] the amalgam is a priori required to live in the corresponding Kac-Moody group. This is rather inconvenient since it gives no intrinsic description of the amalgam. Our result above defines Kac-Moody groups as universal completions of certain abstract amalgams hence giving concrete presentations for those groups.

Note that Theorem 1.1 together with Corollary 1.2 provides an alternative proof of the classification of sound Moufang foundations with connected simply-laced triangle-free Dynkin diagram as given in [28, Sect. 6.5].

We can refine Corollary 1.2 as follows. See Sect. 5.2 for the exact definitions.

Corollary 1.3

Let $\varGamma$ be a connected simply laced Dynkin diagram with no triangles and $k$ a field with at least 4 elements. Any locally split Kac-Moody group over $k$ with diagram $\varGamma$ is a central quotient of a twist $G_\varGamma ^\delta (k)$ of the Curtis-Tits amalgam inside the corresponding split Kac-Moody group. Moreover any two twists are equivalent if and only if they correspond to the same homomorphism $\varPhi :\pi (\varGamma , i_0)\rightarrow {\mathbb Z}_2\times \mathrm{Aut}(k)$.

For a definition of the twist $G_\varGamma ^\delta (k)$ see Sect. 6.

Conceivably Corollary 3 above can be proved directly from results of Tits and Mühlherr in [28] and [17]. Indeed in [17] a bijective correspondence is established between Moufang foundations and twin-buildings and in [28] whereas [28, Sect. 6.5] contains the correspondence between Moufang foundations and homomorphisms $\pi _1(\varGamma ,i_0)\rightarrow \mathrm{Aut}(k)$. However, to our knowledge no explicit correspondence between Curtis-Tits amalgams and Moufang foundations exists in the literature. As for Corollary 1.2, it provides a group theoretic definition of Kac-Moody groups that does not make a reference to Moufang foundations or other data that are a priori associated to twin-buildings. Indeed, in the absence of Theorem 1.1, it is not immediately obvious that different choices of an orientable CT amalgam would give different foundations. See also Corollary 2 for a more precise construction of the amalgams in the spirit of [12] (see the application to Theorem A in loc. cit.).

The paper is organized as follows. In Sect. 2 we review our modification of Bass-Serre theory from [6] and add a few improvements. In Sect. 3 we define Curtis-Tits structures and their morphisms, and prove some general technical lemmas regarding Property (D). In Sect. 4 we prove Theorem 1.1. In Sect. 5 we prove Corollary 1.2 and in Sect. 6 we prove Corollary 1.3.

Note. In the original version of the present paper we developed a technique for classifying amalgams and applied this to the classification of Curtis-Tits amalgams with simply laced Dynkin diagram. By coincidence, the paper [6], which contains a generalization of that classification technique, was published first. In the interest of self-containedness and efficiency, in the present paper we will quickly review the results from loc. cit., add a few improvements, and then focus on their application in the classification of Curtis-Tits structures with simply laced Dynkin diagram.

Acknowledgement. The original version of this paper was written during some wonderful, if very claustrophobic and accident prone three weeks in Birmingham. We thank Irina and Karin for putting up with it all.

The authors would also like to thank the two referees for their excellent comments, which improved the final version and eliminated some ambiguities in the manuscript.

2 Classification of Amalgams Using Graphs of Groups

In this section we review some definitions and results from [6] that are essential to our main results and furnish some additional details. For proofs, examples and applications, we refer to loc. cit.

2.1 Amalgams, Graphs of Groups, and Pointings

Throughout the paper we fix an oriented graph $\overrightarrow{\varGamma }=(I,\overrightarrow{E})$ where for each edge $e\in \overrightarrow{E}$ there is a distinct reverse edge denoted $\bar{e}$. Moreover we denote by $d_0(e)$ the starting node of the oriented edge $e$ and by $d_1(e)$ the end vertex of $e$. Thus $d_0(e)=d_1(\bar{e})$. Morphisms of graphs must preserve these properties.

We shall call the graph admissible if it is connected and has no circuits of length $\le 3$ and we will only consider such graphs.

Definition 2.1

An amalgam over the graph $\overrightarrow{\varGamma }=(I,\overrightarrow{E})$ is a collection ${\fancyscript{G}}=\{G_i,G_e,\varphi _{e}\mid i\in I, e\in \overrightarrow{E}\}$, where $G_i$ and $G_e=G_{\bar{e}}$ are groups and, for each $e$ we have a monomorphism $\varphi _e:G_{d_0(e)}\hookrightarrow G_e$, called an inclusion map. A completion of ${\fancyscript{G}}$ is a group $G$ together with a collection $\phi =\{\phi _e,\phi _i\mid i\in I, e\in \overrightarrow{E}\}$ of homomorphisms $\phi _e:G_e\rightarrow G$, $\phi _i:G_i\rightarrow G$ such that for any $e$ we have $\phi _e\mathbin { \circ }\varphi _e=\phi _{d_0(e)}$. For simplicity we shall write $\bar{G}_e=\varphi _{e}(G_{d_0(e)})\le G_{e}$. The amalgam ${\fancyscript{G}}$ is non-collapsing if it has a non-trivial completion. A completion $(\widetilde{G},\widetilde{\phi })$ is called universal if for any completion $(G,\phi )$ there is a (necessarily unique) surjective group homomorphism $\pi :\widetilde{G}\rightarrow G$ such that $\phi =\pi \mathbin { \circ }\widetilde{\phi }$.

Definition 2.2

A special homomorphism between the two amalgams

$$\begin{aligned} {\fancyscript{G}}^{(1)}(\overrightarrow{\varGamma })&=\{G_i^{(1)},G_e^{(1)},\varphi _{e}^{(1)}\mid i\in I, e\in \overrightarrow{E}\} \text{ and } \\ {\fancyscript{G}}^{(2)}(\overrightarrow{\varGamma })&=\{G_i^{(2)},G_e^{(2)},\varphi _{e}^{(2)}\mid i\in I, e\in \overrightarrow{E}\} \end{aligned}$$

is a map $\phi =\{ \phi _i, \phi _{e}\mid i\in I, e \in \overrightarrow{E}\}$ such that, for all $e\in \overrightarrow{E}$

is a commutative diagram of group homomorphisms. We call $\phi$ an isomorphism of amalgams if $\phi _i$ and $\phi _{e}$ are bijective for all $i,j\in I$, and $\phi ^{-1}$ is a homomorphism of amalgams.

Remark 2.3

One could define a more general notion of homomorphism of amalgams involving automorphisms of the graph $\varGamma$. We have chosen not to do this and therefore fix a labeling $I$ of the vertices of $\varGamma$ and the corresponding groups in ${\fancyscript{G}}$ throughout.

Definition 2.4

Consider an amalgam ${\fancyscript{G}}_0=\{G_i,G_{e},\psi _{e}\mid i\in I, e\in \overrightarrow{E}\}$ over $\overrightarrow{\varGamma }=(I,\overrightarrow{E})$. An amalgam of type ${\fancyscript{G}}_{0}$ is an amalgam over $\overrightarrow{\varGamma }$ with the same groups $G_i, G_{e}$ and where, for every $e\in \overrightarrow{E}$, the inclusion map $\varphi _{e}$ is possibly different from $\psi _{e}$ except that it has the same image $\bar{G}_{e}$.

For the rest of this section, ${\fancyscript{G}}_0$ will be a fixed amalgam over $\overrightarrow{\varGamma }=(I,\overrightarrow{E})$. We shall therefore simply write ${\fancyscript{G}}_0=\{G_i,G_e,\psi _{e}\}$ tacitly understanding that $i$ is taken over all elements in $I$ and $e$ is taken over all elements of $\overrightarrow{E}$. We shall adopt a similar shorthand for amalgams ${\fancyscript{G}}$ of type ${\fancyscript{G}}_0$.

Given a collection of subgroups $G_1,\ldots ,G_k$ of the group $G$, we define $\mathrm{Aut}_G(G_1,\ldots ,G_k)$ to be the subgroup of $\mathrm{Aut}(G)$ stabilizing each $G_i$. Given a monomorphism of groups $\phi :G\rightarrow H$, there is a corresponding homomorphism $\mathrm{ad}(\phi ):\mathrm{Aut}_H(\phi (G))\rightarrow \mathrm{Aut}(G)$ such that for any $a\in \mathrm{Aut}_H(\phi (G))$ we have $\mathrm{ad}(\phi )(a)=\phi ^{-1}\mathbin { \circ } a\mathbin { \circ } \phi$.

Lemma 2.5

Let $f\in \overrightarrow{E}$ and suppose that the set map

$$\begin{aligned} \mathrm{Aut}_{G_f}(\bar{G}_{f},\bar{G}_{\bar{f}})\rightarrow \mathrm{Aut}(\bar{G}_{f})\times \mathrm{Aut}(\bar{G}_{\bar{f}}) \end{aligned}$$

given by restriction is onto. Then the isomorphism class of any amalgam ${\fancyscript{G}}=\{G_i,G_e,\varphi _e\}$ of type ${\fancyscript{G}}_0$ does not depend on $\varphi _f$ or $\varphi _{\bar{f}}$.

Proof

For $i=1,2$, let ${\fancyscript{G}}^{(i)}=(G_i,G_e,\varphi _e^{(i)}\}$ be an amalgam of type ${\fancyscript{G}}_0$. Suppose that $\varphi _e^{(1)}=\varphi _e^{(2)}$ for all $e\in \overrightarrow{E}-\{f,\bar{f}\}$. Then since the groups $\bar{G}_i$ and $\bar{G}_e$ are the same in both amalgams, the compositions $a_{d_0(f)}=\varphi _f^{(2)}\mathbin { \circ }\mathrm{id}_{G_{d_0(f)}}\mathbin { \circ }(\varphi _f^{(1)})^{-1}$ and $a_{d_1(f)}=\varphi _{\bar{f}}^{(2)}\mathbin { \circ }\mathrm{id}_{G_{d_0(\bar{f})}}\mathbin { \circ }(\varphi _{\bar{f}}^{(1)})^{-1}$ are automorphisms of $\bar{G}_{f}$ and $\bar{G}_{\bar{f}}$ respectively. By assumption there exists $a_f\in \mathrm{Aut}_{G_f}(\bar{G}_{f},\bar{G}_{\bar{f}})$ that simultaneously induces $a_{d_0(f)}$ and $a_{d_1(f)}$. It follows that the set $\phi =\{\phi _i,\phi _e\mid i\in I, e\in \overrightarrow{E}\}$ where

$$\begin{aligned} \phi _i=\mathrm{id}_{G_i} \text{ for } \text{ all } i\in I&\text{ and }&\phi _e={\left\{ \begin{array}{ll} a_f &{} \text{ if } e=f\\ \mathrm{id}_{G_e} &{} \text{ else } \end{array}\right. } \end{aligned}$$

is an isomorphism ${\fancyscript{G}}^{(1)}\rightarrow {\fancyscript{G}}^{(2)}$.

Remark 2.6

• (a) The situation of Lemma 2.5 arises for instance if $G_f$ equals $\bar{G}_{f}\oplus \bar{G}_{\bar{f}}$ (internal direct sum) or $\bar{G}_{f} * \bar{G}_{\bar{f}}$ (free product).

• (b) The situation of Lemma 2.5 also arises if $G_f\cong \bar{G}_f\circ \bar{G}_{\bar{f}}$ (central product), and $\bar{G}_f\cong \bar{G}_{\bar{f}}\cong \mathrm{SL}_2(k)$ for some field $k$. Namely, the center of $\mathrm{SL}_2(k)$ equals $\{\pm 1\}$ and any automorphism of $\mathrm{SL}_2(k)$ will fix this. Therefore $\mathrm{Aut}_{G_f}(\bar{G}_{f},\bar{G}_{\bar{f}})\cong \mathrm{Aut}_{\bar{G}_f\times \bar{G}_{\bar{f}}}(\bar{G}_{f},\bar{G}_{\bar{f}})$.

• (c) As a consequence of Lemma 2.5, for the purposes of classifying amalgams of type ${\fancyscript{G}}_0$ up to isomorphism, we can remove any edges $f$ as in that lemma from $\overrightarrow{E}$ and assume that $\varphi _f=\psi _f$.

Definition 2.7

A weak graph of groups is a pair $({\mathbb {A}}, \overrightarrow{\varGamma })$ where $\overrightarrow{\varGamma }$ is a graph as above and ${\mathbb {A}}$ associates to each $i\in I$ a group $A_i$ and to each directed edge $e\in \overrightarrow{E}$ a group $A_e=A_{\bar{e}}$. Moreover, for each edge we have a homomorphism $\alpha _{e}:A_{e}\rightarrow A_{d_0(e)}$. We call the weak graph of groups a graph of groups if each $\alpha _e$ is a monomorphism; these are graphs of groups in the sense of [2] and [21]. Since we shall work with a fixed graph $\overrightarrow{\varGamma }=(I,\overrightarrow{E})$, if we want to specify the groups of $({\mathbb {A}},\overrightarrow{\varGamma })$ we shall write ${\mathbb {A}}=\{A_i,A_e,\alpha _e\}$ rather than ${\mathbb {A}}=\{A_i,A_e,\alpha _e\mid i\in I, e\in \overrightarrow{E}\}$.

Definition 2.8

Consider the reference amalgam ${\fancyscript{G}}_0=\{G_i,G_{e},\psi _{e}\}$. Recall that every amalgam of type ${\fancyscript{G}}_0$ has the same target subgroups $\bar{G}_e$. Define a weak graph of groups as follows. For each $e\in \overrightarrow{E}$ we take $A_{d_0(e)}=\mathrm{Aut}(G_{d_0(e)})$, $A_{e}=\mathrm{Aut}_{G_{e}}(\bar{G}_{e}, \bar{G}_{\bar{e}})$ and $\alpha _{e}=\mathrm{ad}(\psi _{e})$. The resulting weak graph of groups will be denoted by $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ and will be called the reference graph of groups.

Remark 2.9

Since the maps $\alpha _e$ in Definition 2.8 are not necessarily monomorphic, so $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ is in general not a proper graph of groups in the sense of [2, 21] or in the more general language of complexes of groups as described and developed in [9, Part III.C].

Definition 2.10

Given graphs of groups $({\mathbb {A}}^{(k)},\overrightarrow{\varGamma })$ for $k=1,2$, an inner morphism is a collection $\phi =\{\phi _i,\phi _{e}\mid i\in I, e\in \overrightarrow{E}\}$ of group homomorphisms $\phi _i:A_i^{(1)}\rightarrow A^{(2)}_{i}$ and $\phi _{e}:A_{e}^{(1)}\rightarrow A^{(2)}_{e}$ so that for each $e\in \overrightarrow{E}$ there exists an element $\delta _{e}\in A_{d_0(e)}^{(2)}$ so that

$$ \phi _{d_0(e)}\mathbin { \circ } \alpha _{e}^{(1)}=\mathrm{ad}(\delta _{e}^{-1})\mathbin { \circ }\alpha _{e}^{(2)}\mathbin { \circ } \phi _{e}. $$

Here $\mathrm{ad}(x)(y)=x^{-1}yx$. We call an inner morphism central if $\delta _e=1$ for all $e\in \overrightarrow{E}$.

Definition 2.11

Let $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ be a weak graph of groups, a pointing is a pair $(({\mathbb {A}},\overrightarrow{\varGamma }),\delta )$, where $\delta =\{\delta _{e}\mid e\in \overrightarrow{E}\}$ is a collection of elements $\delta _{e}\in A_{d_0(e)}$ and $({\mathbb {A}},\overrightarrow{\varGamma })$ is a weak graph of groups obtained from $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ by setting $\alpha '_{e}=\mathrm{ad}(\delta _{e}^{-1})\mathbin { \circ }\alpha _{e}$, for each $e\in \overrightarrow{E}$.

Lemma 2.12

For any pointing $(({\mathbb {A}},\overrightarrow{\varGamma }),\delta )$ of $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ the weak graphs of groups $({\mathbb {A}},\overrightarrow{\varGamma })$ and $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ are isomorphic.

Definition 2.13

An isomorphism between pointings $(({\mathbb {A}},\overrightarrow{\varGamma }),\delta ^{(k)})$ ($k=1,2$) of a weak graph of groups $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ is an inner isomorphism $\phi$ of $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ such that there exist ${m}_i\in A_i$ and ${m}_{e}\in A_{e}$ so that ${m}_{e}= {m}_{\bar{e}}$ and with $\phi _i=\mathrm{ad}({m}_i)$ and $\phi _{e}=\mathrm{ad}({m}_{e})$ for each $i\in I, e\in \overrightarrow{E}$ and we have

$$\begin{aligned} \delta _{e}^{(1)}\alpha _{e}({m}_{e})= {m}_{d_0(e)}\delta _{e}^{(2)}. \end{aligned}$$

(1)

We will say that $\{{m}_{e}, {m}_i\mid i\in I, e\in \overrightarrow{E}\}$ induces the isomorphism.

Definition 2.14

If ${\fancyscript{G}}=\{G_i,G_{e}, \varphi _{e}\}$ is an amalgam of type ${\fancyscript{G}}_0$, we define an associated pointing $(({\mathbb {A}},\overrightarrow{\varGamma }),\delta ^{\fancyscript{G}})$ of ${\mathbb {A}}_0$ as follows. If $e\in \overrightarrow{E}$ then $\delta ^{\fancyscript{G}}_{e}=\varphi _{e}^{-1}\mathbin { \circ }\psi _{e}$ where of course by $\varphi _{e}^{-1}$ we mean the inverse of the map $\varphi _{e}:G_{d_0(e)}\rightarrow \bar{G}_{e}$.

Conversely if $(({\mathbb {A}}, \overrightarrow{\varGamma }),\delta )$ is a pointing of ${\mathbb {A}}_0$ then we define an amalgam ${\fancyscript{G}}^\delta =\{G_i,G_e,\varphi ^\delta _{e}\}$ of type ${\fancyscript{G}}_0$ via $\varphi ^\delta _{e}=\psi _{e}\delta _e^{-1}$.

Theorem 2.15

[6, Theorem 1] The correspondence ${\fancyscript{G}}\mapsto (({\mathbb {A}},\overrightarrow{\varGamma }),\delta ^{\fancyscript{G}})$ with inverse $(({\mathbb {A}},\overrightarrow{\varGamma }),\delta )\mapsto {\fancyscript{G}}^\delta$ yields a bijection between special isomorphism classes of amalgams of type ${\fancyscript{G}}_0$ and isomorphism classes of pointings of ${\mathbb {A}}_0$.

2.2 Amalgams and Fundamental Groups

Definition 2.16

Call the weak graph of groups $({\mathbb {A}},\overrightarrow{\varGamma })$ bijective if each $\alpha _e$ is bijective. Note that this means in particular that $({\mathbb {A}},\overrightarrow{\varGamma })$ is a graph of groups in the sense of [2, 21].

In our classification of Curtis-Tits structures with admissible Dynkin diagram we shall demonstrate that we can restrict ourselves to pointings of bijective graphs of groups. In anticipation of this, in this subsection we make the following assumption:

(B):

${\mathbb {A}}_0$ is bijective.

Since ${\mathbb {A}}_0$ is a graph of groups in the more restricted sense of [2] we can use Definitions 2.17 and 2.18 introduced there.

Definition 2.17

For a given graph of groups $({\mathbb {A}}, \overrightarrow{\varGamma })$ we define the path group as follows

$$ \pi ({\mathbb {A}})=((*_{i\in I} A_i )* F(\overrightarrow{E}))/ R$$

where $F(\overrightarrow{E})$ is the free group on the set $\overrightarrow{E}$, $*$ denotes free product and $R$ is the following set of relations: for any $e\in \overrightarrow{E}$, we have

$$\begin{aligned} \begin{array}{rll} e\bar{e} =&{}\mathrm{id} &{} \qquad \qquad \text{ and } \\ e\cdot \alpha _{\bar{e}}(a)\cdot \bar{e} =&{}\alpha _e(a) &{} \qquad \qquad \text{ for } \text{ any } a \in A_e. \end{array} \end{aligned}$$

(2)

Definition 2.18

Given a graph of groups $({\mathbb {A}}, \overrightarrow{\varGamma })$, a path of length $n$ from $i_1$ to $i_{n+1}$ in ${\mathbb {A}}$ is a sequence $\gamma =(a_1, e_1, a_2, \ldots , e_{n}, a_{n+1})$, where $e_1, \ldots , e_n$ is an edge path in $\overrightarrow{\varGamma }$ with vertex sequence $i_1, \ldots , i_{n+1}$ and $a_k \in A_{i_k}$ for each $k=1,\ldots ,n+1$. We call $\gamma$ reduced if it has no returns. Since $({\mathbb {A}},\overrightarrow{\varGamma })$ is bijective this is equivalent to saying that $e_{i+1}\ne \bar{e}_i$ for any $i=1,\ldots ,{n-1}$. The path $\gamma$ defines an element $|\gamma |=a_1\cdot e_1 \cdot a_2 \cdots e_{n} \cdot a_{n+1} \in \pi ({\mathbb {A}})$. We denote by $\pi [i,j]$ the collection of elements $|\gamma |$, where $\gamma$ runs through all paths from $i$ to $j$ in ${\mathbb {A}}$. Concatenation induces a group operation on $\pi ({\mathbb {A}}, i_0)=\pi [i_0,i_0]$ and we call this group the fundamental group of ${\mathbb {A}}$ with base point $i_0$.

Remark 2.19

One verifies easily that we can naturally identify

$$\begin{aligned} \pi _1(\overrightarrow{\varGamma },i_0)=\pi ({\mathbb {A}},i_0)\cap \langle e\mid e\in \overrightarrow{E}\rangle , \end{aligned}$$

where the former denotes the fundamental group of the graph $\overrightarrow{\varGamma }$ with base point $i_0$.

Consider the bijective graph of groups $({\mathbb {A}}_0,\overrightarrow{\varGamma })$ and let $\eta =e_1,\ldots ,e_n$ be a path from $l$ to $m$ in $\overrightarrow{\varGamma }$. Define $\beta _\eta :A_l\rightarrow A_m$ by setting $\beta _\eta (a)=\alpha _{\bar{e}_n}\mathbin { \circ } \alpha _{e_n}^{-1}\mathbin { \circ } \cdots \mathbin { \circ }\alpha _{\bar{e}_1}\mathbin { \circ } \alpha _{e_1}^{-1}(a)$, for each $a\in A_l$.

Lemma 2.20

Any element $|\gamma | \in \pi ({\mathbb {A}}_0, i_0)$ can be uniquely realized as $e_1e_2\cdots e_n g$ for some edge path $\eta =e_1,\ldots ,e_n$ with $d_0(e_1)=i_0=d_1(e_n)$ and $g \in A_{i_0}$. More precisely, if $\gamma = (d_1,e_1, d_2,\ldots , d_{n}, e_n, d_{n+1})$ with $d_k\in \mathbf{A}_{d_0(e_k)}$ for $k=1,\ldots ,n$, and $d_{n+1}\in \mathbf{A}_{i_0}$, then we have

$$\begin{aligned} g= \beta _{\bar{\eta }}(d_1)\cdots \beta _{\bar{e}_n\bar{e}_{n-1}}(d_{n-1})\cdot \beta _{\bar{e}_n}(d_n)d_{n+1}.\end{aligned}$$

Proof

The first part is a special case of Corollary 1.13 in [2] since all maps $\alpha _e$ are surjective. The second part follows from the relations in Definition 2.17 and the definition of $\beta _\eta$ above.

Definition 2.21

If $(({\mathbb {A}}, \overrightarrow{\varGamma }), \delta )$ is a pointing of the graph of groups $({\mathbb {A}}_0, \overrightarrow{\varGamma })$ then any edge path $\eta =e_1, \ldots , e_n$ in $\overrightarrow{\varGamma }$ gives rise to a path in ${\mathbb {A}}$ via $\eta \mapsto \eta _{\delta }=\delta _{e_1} e_1 \delta _{\bar{e}_1}^{-1}\delta _{e_2} \cdots e_{n-1}\delta _{\bar{e}_{n-1}}^{-1}\delta _{e_n}e_n \delta _{\bar{e}_n}^{-1}$. Fixing a base point $i_0$, the map $\eta \mapsto |\eta _\delta |$ restricts to a monomorphism

$$\begin{aligned} \varPhi _\delta :\pi _1( \varGamma , i_0) \rightarrow \pi ({\mathbb {A}}_0,i_0), \end{aligned}$$

where $\pi _1(\varGamma ,i_0)$ denotes the fundamental group of $\varGamma$ with base point $i_0$. The image of this map is called the fundamental group of the pointing and denoted by $\pi ({\mathbb {A}},i_0,\delta )$.

Define the map $\varPi _\delta :\pi _1(\overrightarrow{\varGamma },i_0)\rightarrow A_{i_0}$ as follows: given an edge path $\eta =e_1,\ldots ,e_n$ with $|\eta |\in \pi _1(\overrightarrow{\varGamma },i_0)$, let $\gamma =\eta _\delta$ and let $\varPi _\delta (\eta )=g$, where $\gamma$ and $g$ are related as in Lemma 2.20. Concretely:

$$\begin{aligned} \varPi _\delta (\eta )=\beta _{\bar{\eta }}(\delta _{e_1})\cdots \beta _{\bar{e}_n\bar{e}_{n-1}}(\delta _{\bar{e}_{n-2}}^{-1}\delta _{e_{n-1}})\cdot \beta _{\bar{e}_n}(\delta _{\bar{e}_{n-1}}^{-1}\delta _{e_n})\cdot \delta _{\bar{e}_n}^{-1}. \end{aligned}$$

Lemma 2.22

We have $\pi ({\mathbb {A}}_0, i_0)\cong \pi (\overrightarrow{\varGamma },i_0)\ltimes A_{i_0}$. Let $H=\{\eta _j\}_{j\in J}$ be a base for the free group $\pi _1(\overrightarrow{\varGamma },i_0)$. Then, both $\varPhi _\delta$ and its image $\pi _1({\mathbb {A}},i_0,\delta )$ are uniquely determined by the assignment $\eta \mapsto \varPi _\delta (\eta )$.

Proof

Clearly as subgroups of $\pi _1({\mathbb {A}},i_0)$ we have $A_{i_0}\cap \pi _1(\overrightarrow{\varGamma },i_0)=\{1\}$. From Lemma 2.20 it follows in particular, that $\pi _1(\overrightarrow{\varGamma },i_0)$ normalizes $A_{i_0}$. This proves the first statement. Now given the base $H$, and the assignment $\eta \mapsto \varPi _\delta (\eta )$, using Lemma 2.20 we see that $\varPhi _\delta (\eta )=\eta \varPi _\delta (\eta )$ and $\pi _1({\mathbb {A}},i_0,\delta )=\langle \eta \varPi _\delta (\eta )\mid \eta \in H\rangle \le \pi _1({\mathbb {A}},i_0)$. Conversely, given $\eta \in H$ since the composition $\pi _1({\mathbb {A}},i_0,\delta )\hookrightarrow \pi _1(\overrightarrow{\varGamma },i_0)\ltimes A_{i_0}\rightarrow \pi _1(\overrightarrow{\varGamma },i_0)$ yields an isomorphism, there is a unique $a\in A_{i_0}$ such that $\eta a\in \pi _1({\mathbb {A}},i_0,\delta )$. Thus, we must have $\eta \mapsto \varPi _\delta (\eta )=a$.

Corollary 2.23

The fundamental groups of pointings are bijectively parametrized by the homomorphisms $\pi _1(\overrightarrow{\varGamma },i_0)\rightarrow A_{i_0}$.

Proof

The data $\{(\eta ,\varPi _\delta (\eta ))\mid \eta \in \pi _1(\overrightarrow{\varGamma },i_0)\}$ from Lemma 2.22 bijectively parametrizes such homomorphisms.

The following theorem appears as Theorem 3.7 in [6].

Theorem 2.24

Two pointings of ${\mathbb {A}}_{0}$ are isomorphic if and only if they have the same fundamental group.

Remark 2.25

Note that all such fundamental groups are isomorphic to the fundamental group $\pi _1(\varGamma ,i_0)$, but not all of them are the same.

Corollary 2.26

The isomorphism classes of amalgams of type ${\fancyscript{G}}_0$ are in bijective correspondence with the homomorphisms $\pi _1(\overrightarrow{\varGamma },i_0)\rightarrow A_{i_0}$.

Proof

This follows from Theorems 2.15 and 2.24, and Corollary 2.23.

3 CT-Structures

In this section we introduce the notion of a CT-structure over a commutative field and define its category. Throughout the paper $k$ will be a commutative field, which after Lemma 3.5 will be assumed to have at least four elements.

3.1 CT-Structures and Standard Pairs

Definition 3.1

Let $\varGamma$ be a Dynkin diagram over a set $I$ and let $K(I)$ be the complete graph on $I$. Let $\overrightarrow{ K(I)}=(I,\overrightarrow{F})$ be the oriented graph associated to $K(I)$ and let $\overrightarrow{\varGamma }=(I,\overrightarrow{E})$ be the oriented graph associated to $\varGamma$; that is, $e\in \overrightarrow{E}$ if and only if the subdiagram of $\varGamma$ induced on $e$ is not of type $A_1\times A_1$.

A Curtis-Tits (CT) structure over $k$ with Dynkin diagram $\varGamma$ is an amalgam ${\fancyscript{G}}=\{G_i,G_e,\varphi _e\mid i\in I, e\in \overrightarrow{F}\}$ over $\overrightarrow{ K(I)}$ such that, for every $e\in \overrightarrow{F}$, $\{\bar{G}_e,G_e,\bar{G}_{\bar{e}}\}$ is a Curtis-Tits amalgam with respect to a $BN$-pair of type $\varGamma _e$ for $G_e$. Here $\varGamma _e$ denotes the subdiagram of $\varGamma$ induced on the edge $e$.

Definition 3.2

A simply laced Dynkin diagram over the set $I$ is a simple graph $\varGamma =(I,E)$. That is, $\varGamma$ has vertex set $I$, and an edge set $E$ that contains no loops or double edges.

In the present paper we shall restrict ourselves to the case where $\varGamma$ is simply-laced. To specify the Curtis-Tits amalgam in each rank-$2$ group of a CT-structure over $\varGamma$, we make use of standard pairs.

Definition 3.3

Let $V$ be a vector space of dimension $3$ over $k$. We call $(S_1,S_2)$ a standard pair for $S=\mathrm{SL}(V)$ if there are decompositions $V=U_i\oplus V_i$, $i=1,2$, with $\dim (U_i)=1$ and $\dim (V_i)=2$ such that $U_1\subseteq V_2$ and $U_2\subseteq V_1$ and $S_i$ is the subgroup of elements in $S$ that centralize $U_i$ and preserve $V_i$.

One also calls $S_1$ a standard complement of $S_2$ and vice-versa. We set $D_1= N_{S_1}(S_2)$ and $D_2=N_{S_2}(S_1)$. A simple calculation shows that $D_i$ is a maximal torus in $S_i$, for $i=1,2$. In general if $G\cong \mathrm{SL}_3(k)$, then $(G_1,G_2)$ is a standard pair for $G$ if there is an isomorphism $\psi :G\rightarrow S$ such that $\psi (G_i)=S_i$ for $i=1,2$.

Definition 3.4

Given a standard pair $(S_1,S_2)$, a standard basis for $(S_1, S_2)$ is an ordered basis $\mathcal {V}=(v_1,v_2,v_3)$ of $V$ such that $V_1=\langle v_1,v_2\rangle$, $U_1= \langle v_3\rangle$, $U_2=\langle v_1\rangle$, and $V_2=\langle v_2,v_3\rangle$.

Identifying $S$ with $\mathrm{SL}_3(k)$ via its left action on $V$ with respect to $\mathcal {V}$, yields

$$\begin{aligned} S_1=\left\{ \left. \begin{pmatrix} A &{} 0 \\ 0 &{} 1\end{pmatrix} \right| A \in \mathrm{SL}_{2}(k)\right\}&\text{ and }&S_2=\left\{ \left. \begin{pmatrix} 1 &{} 0 \\ 0 &{} A\end{pmatrix} \right| A \in \mathrm{SL}_2(k)\right\} \\ \text{ so } \text{ that }&\\ D_1 =\left\{ \left. \begin{pmatrix} a &{} 0 &{} 0 \\ 0 &{} a^{-1} &{} 0\\ 0 &{} 0 &{} 1 \end{pmatrix} \right| a \in k^*\right\}&\text{ and }&D_2=\left\{ \left. \begin{pmatrix} 1 &{} 0 &{} 0\\ 0 &{} a &{} 0\\ 0 &{} 0 &{} a^{-1} \end{pmatrix}\right| a \in k^*\right\} . \end{aligned}$$

Lemma 3.5

Let $S_1$ and $S_2$ be a standard pair for $S=\mathrm{SL}_3(k)$, where $k$ has at least four elements.

• (a) The group $D_1$ determines a standard basis $\mathcal {V}=(v_1,v_2,v_3)$ up to a permutation of $\{v_1,v_2\}$ and a linear diagonal transformation with respect to $\mathcal {V}$.

• (b) The pair $(D_1,D_2)$ and hence, a fortiori, the pair $(S_1,S_2)$ determines a standard basis $\mathcal {V}$ for $(S_1,S_2)$ up to a linear diagonal transformation with respect to $\mathcal {V}$.

• (c) The group $S_1$ has exactly one standard complement $S_2'\ne S_2$ normalized by $D_1$.

Proof

Since $k$ has at least four elements, $D_1$ uniquely determines three $1$-dimensional eigenspaces and $S_1$ fixes all vectors in exactly one of these eigenspaces, say $E_1$. In the notation above, these are $E_1=U_1$, $U_2$ and $V_1\cap V_2$. Any standard basis $\mathcal {V}=(v_1,v_2,v_3)$ must satisfy $U_2=\langle v_1\rangle$, $V_1\cap V_2=\langle v_2\rangle$, and $U_1=\langle v_3\rangle$. Thus (3.5) and hence (3.5) follow. To see (3.5), note that any standard complement $S_2$ to $S_1$ that is normalized by $D_1$ is completely determined by the eigenspace $E \ne E_1$ that it fixes vector-wise. As we saw, there are two choices.

We will need the following lemma.

Lemma 3.6

With the notations above, $D_{1}=C_{S_{1}}(D_2)$ and $D_{2}=C_{S_{2}}(D_1)$. Moreover, $D_{2}$ is the only torus in $S_{2}$ that is normalized by $D_{1}$.

Proof

Note that if $T$ is a torus in $S_{2}$ then $N_{S}(T)$ is the set of monomial matrices so $N_{S_{1}}(T)$ only contains one torus which is $C_{S_{1}}(T)$. The conclusion follows.

Definition 3.7

Let $\varGamma =(I,E)$ be a simply laced Dynkin diagram. A Curtis-Tits structure over $\varGamma$ is a non-collapsing amalgam ${\fancyscript{G}}(\varGamma )=\{G_{i},G_e,\varphi _e\mid i\in I,e\in \overrightarrow{F} \}$ over $\overrightarrow{ K(I)}$ such that

• (CT1) for each vertex $i$, $G_i = \mathrm{SL}_2(k)=\mathrm{SL}(W_i)$, for some $2$-dimensional vector space $W_i$ over $k$, and for each edge $e\in \overrightarrow{F}$,

  $$\begin{aligned} G_e\cong {\left\{ \begin{array}{ll} \mathrm{SL}(V_e) &{} \text{ if } \ e\in \overrightarrow{E}\\ \bar{G}_e\circ \bar{G}_{\bar{e}} &{} \text{ if } \ e\not \in \overrightarrow{E},\end{array}\right. } \end{aligned}$$

  where $V_e=V_{\bar{e}}$ is a 3-dimensional vector space over $k$, $\bar{G}_e=\varphi _e(G_{d_0(e)})$, and $\circ$ denotes central product,

• (CT2) if $e\in \overrightarrow{E}$ then $(\bar{G}_{e}, \bar{G}_{\bar{e}})$ is a standard pair in $G_e$.

Remark 3.8

In view of Remark 2.6 for the purposes of classifying CT-structures with simply-laced Dynkin diagram, we may in fact classify the subamalgams defined over $\overrightarrow{\varGamma }$ only. In practice this means that, for $f\in \overrightarrow{F}-\overrightarrow{E}$, we can let $\varphi _f$ be the same in every amalgam of this type.

Recall that we shall only consider admissible Dynkin diagrams, that is, graphs that are connected and have no circuits of length $\le 3$.

From now on $\varGamma =(I,E)$ will be an admissible Dynkin diagram and our reference amalgam ${\fancyscript{G}}_0={\fancyscript{G}}(\varGamma )=\{G_i,G_e,\psi _e\mid i\in I,e\in \overrightarrow{F}\}$ will be a non-collapsing Curtis-Tits structure over $\varGamma$.

We now focus on the following subgroups:

Definition 3.9

For any $e\in \overrightarrow{E}$, let

$$\begin{aligned} D_e=N_{G_e}(\bar{G}_{\bar{e}})\cap \bar{G}_e \end{aligned}$$

As noted in Definition 3.3, $D_e$ is a torus in $\bar{G}_e\le G_e$. By Lemma 3.6 $D_e$ is the only torus in $\bar{G}_e$ normalized by $D_{\bar{e}}$.

Lemma 3.10

If $e\in \overrightarrow{E}$, then $D_e$ and $D_{\bar{e}}$ are contained in a unique common maximal torus $D_{e,\bar{e}}$ of $G_e$.

Proof

Clearly in any completion of the amalgam, both $D_e$ and $D_{\bar{e}}$ normalize $\bar{G}_e$ and $\bar{ G}_{\bar{e}}$ so we have $D_e, D_{\bar{e}}\le N_{G_e}(\bar{G}_e)\cap N_{G_e}(\bar{G}_{\bar{e}})=D_{e,\bar{e}}$, which is the required maximal torus.

Definition 3.11

(property (D)) Let ${\fancyscript{G}}=\{G_i,G_e,\varphi _e\mid i\in I,e\in \overrightarrow{F}\}$ be a Curtis-Tits structure over $\varGamma$. We say that ${\fancyscript{G}}$ has property (D) if, for each $i\in I$ there is a subgroup $D_i\le G_i$ such that for any pair of edges $e,f\in \overrightarrow{E}$ with $d_0(f)=i=d_0(e)$ and any completion $(G,\phi )$, we have a commutative diagram of isomorphisms

Property (D) is equivalent to requiring that for all edges $e,f\in \overrightarrow{E}$ with $d_0(f)=d_0(e)$ we have an isomorphism $\varphi _{f}\mathbin { \circ }\varphi _e^{-1}:D_e\rightarrow D_{f}$.

That ${\fancyscript{G}}_0$ is non-collapsing has the following consequence.

Lemma 3.12

Let ${\fancyscript{G}}=\{G_i,G_e,\varphi _e\}$ be a CT-structure with admissible Dynkin diagram $\varGamma$. Suppose that $\varphi _e$ is injective for every $e\in \overrightarrow{E}$. Then, ${\fancyscript{G}}$ has property (D).

Proof

(See also [15]) Let $e,f\in \overrightarrow{E}$ and $i\in I$ be any two edges with $d_0({f})=i=d_0(e)$. By Definition 3.3 $\phi _{\bar{f}}(D_{\bar{f}})$ normalizes $\phi _{f}(\bar{G}_{f})=\phi _i(G_i)=\phi _e(\bar{G}_e)$. It follows from the fact that the nodes $d_0(\bar{f})$ and $d_0(\bar{e})$ are not connected in $\varGamma$ that $\phi _{\bar{f}}(D_{\bar{f}})$ commutes with $\phi _{\bar{e}}(D_{\bar{e}})$. Thus, $\phi _{\bar{f}}(D_{\bar{f}})$ normalizes $\phi _e(\bar{G}_e)$, and hence $\phi _e(G_e)$, while centralizing $\phi _{\bar{e}}(\bar{G}_{\bar{e}})$. Hence, by Lemma 3.6, $\phi _{\bar{f}}(D_{\bar{f}})$ normalizes the torus $\phi _e(D_e)=C_{\phi _e(\bar{G}_e)}(\phi _{\bar{e}}(D_{\bar{e}}))$ of $\phi _e(\bar{G}_e)=\phi _{{f}}(\bar{G}_{{f}})$. By the second part of Lemma 3.6 $\phi _{{f}}(D_{{f}})$ is the only torus in $\phi _{{f}}(G_{{f}})$ normalized by $\phi _{\bar{f}}(D_{\bar{f}})$ and so $\phi _e(D_e)=\phi _{{f}}(D_{{f}})$. The diagram is now commutative since $(G,\phi )$ is a completion of ${\fancyscript{G}}_0$.

From now on we shall only consider Curtis-Tits amalgams with property (D). This allows us to make the following definition.

Definition 3.13

For any $i\in I$, let $e\in \overrightarrow{E}$ be such that $d_0(e)=i$ ($e$ exists since $\varGamma$ is connected). Let $D_i$ be the torus of $G_i$ such that $\varphi _e(D_i)=D_e$ (as in Definition 3.9). Since ${\fancyscript{G}}$ has property (D), $D_i$ does not depend on the choice of $e$.

Moreover, for $e\in \overrightarrow{E}$, we let $D_{e,\bar{e}}$ be the maximal torus generated by $D_e$ and $D_{\bar{e}}$ in $G_e$ (cf. Lemma 3.10).

In case $f\in \overrightarrow{F}-\overrightarrow{E}$, we set

$$\begin{aligned} D_f=\varphi _f(D_{d_0(f)})&\qquad \text{ and }&D_{f,\bar{f}}= D_f D_{\bar{f}}\le G_f. \end{aligned}$$

We collect these groups in a set

$$\begin{aligned} {\fancyscript{D}}({\fancyscript{G}})&=\{D_i,D_f,D_{f,\bar{f}}\mid i\in I, f\in \overrightarrow{F}\}. \end{aligned}$$

Note that $D_e$ and $D_{e,\bar{e}}$ ($e\in \overrightarrow{E}$) only depend on the type of ${\fancyscript{G}}$, $D_i$ ($i\in I$), and hence $D_f$ and $D_{f,\bar{f}}$ ($f\in \overrightarrow{E}-\overrightarrow{F}$) depend, in addition, on the particular connecting maps $\varphi _e$.

Definition 3.14

Note that a torus in $\mathrm{SL}_{2}(k)$ uniquely determines a pair of opposite root groups $X^{+}$ and $X^{-}$. We now choose one root group $X^+_{i}$ normalized by the torus $D_{i}$ of $G_{i}$ for each $i$. We call the CT-structure ${\fancyscript{G}}_0$ having property (D) an orientable Curtis-Tits structure if it admits a system $\{X^+_i\mid i\in I\}$ of root groups as above such that for any $f\in \overrightarrow{F}$ with $d_0(f)=i$ and $d_1(f)=j$, the groups $\varphi _f (X^+_i)$ and $\varphi _{\bar{f}}(X^+_j)$ are contained in a common Borel subgroup $B^+_f=B^+_{\bar{f}}$ of $G_f=G_{\bar{f}}$.

Lemma 3.15

With the notation of Definition 3.14, assume that ${\fancyscript{G}}$ is an orientable CT-structure. Suppose that $\{X_i^+\mid i\in I\}$ is chosen. For each $i\in I$ select the negative root group $X_i^-$. Then, for any $e\in \overrightarrow{E}$, $X_{d_0(e)}^-$ and $X_{d_1(e)}^-$ are contained in a common Borel subgroup $B_e^-$ that is opposite to $B_e^+$.

Proof

Since ${\fancyscript{G}}$ is orientable, it has property (D). The tori $D_e$ and $D_{\bar{e}}$ determine a unique torus in $G_e$ hence a unique apartment of the building associated to $\varDelta _e=(G_e,B_e^+, N_{G_e}(D_{e,\bar{e}}))$. The torus $D_e$ (resp. $D_{\bar{e}}$) determines a pair of opposite panels of $\varDelta _e$. These pairs of opposite panels intersect in a pair of opposite chambers. One corresponds to $B_e^+$ and the other to $B_{\bar{e}}^+=B_e^-$. The conclusion follows.

3.2 Automorphisms of ${\fancyscript{G}}(A_2)$

Let $V$ be a (left) vector space of dimension $n$ over $k$. Let $G=\mathrm{SL}(V)$ act on $V$ as the matrix group $\mathrm{SL}_n(k)$ with respect to some fixed basis $\mathcal {V}=\{v_i\mid i=1,2,\ldots ,n\}$. Let $\omega \in \mathrm{Aut}(\mathrm{SL}_n(k))$ be the automorphism given by

$$\begin{aligned} A\mapsto {}^tA^{-1} \end{aligned}$$

where ${}^tA$ denotes the transpose of $A$.

Let $\varPhi =\{(i,j)\mid 1\le i\ne j\le n\}$. For any $(i,j)\in \varPhi$ and $\lambda \in k$, we define the root group $X_{i,j}=\{X_{i,j}(\lambda )\mid \lambda \in k\}$, where $X_{i,j}(\lambda )$ acts as

$$\begin{aligned} \begin{array}{ll} v_j\mapsto v_j+\lambda v_i &{} \qquad \qquad \text{ and } \\ v_k\mapsto v_k &{} \qquad \qquad \text{ for } \text{ all } k\ne j. \end{array} \end{aligned}$$

Let $\varPhi _+=\{(i,j)\in \varPhi \mid i<j\}$ and $\varPhi _-=\{(i,j)\in \varPhi \mid j<i\}$. We call $X_{i,j}$ positive if $(i,j)\in \varPhi _+$ and negative otherwise. Let $H$ be the torus of diagonal matrices in $\mathrm{SL}_n(k)$ and for $\varepsilon \in \{+,-\}$, let $X_\varepsilon =\langle X_{i,j}\mid (i,j)\in \varPhi _\varepsilon \rangle$ and $B_\varepsilon = H\ltimes X_\varepsilon$.

Let $\varGamma L_n(k)$ be the group of all semilinear automorphisms of the vector space $V$ and let $P\varGamma L_n(k)=\varGamma L_n(k)/Z(\varGamma L_n(k))$. Then $\varGamma L_n(k)\cong \mathrm{GL}_n(k)\rtimes \mathrm{Aut}(k)$, where we view $t\in \mathrm{Aut}(k)$ as an element of $\varGamma L_n(k)$ by setting $((a_{i,j})_{i,j=1}^n)^t=(a_{i,j}^t)_{i,j=1}^n$. The automorphism group of $\mathrm{SL}_n(k)$ can be expressed using $P\varGamma L_n(k)$ and $\omega$ as follows [20].

Lemma 3.16

$$\begin{aligned} \mathrm{Aut}(\mathrm{SL}_n(k))={\left\{ \begin{array}{ll} P\varGamma L_n(k) &{} \text{ if } n=2;\\ P\varGamma L_n(k)\rtimes \langle \omega \rangle &{} \text{ if } n\ge 3.\end{array}\right. } \end{aligned}$$

Remark 3.17

Note that the actions of $\mathrm{Aut}(k)$ and $\omega$ are given with respect to the basis $\mathcal {V}$ of $V$.

4 Classification of CT-Structures with Simply-Laced Diagram

Let ${\fancyscript{G}}$ be a Curtis-Tits structure with property (D) over a simply-laced diagram $\varGamma =(I,E)$. This is really an amalgam ${\fancyscript{G}}=\{G_i,G_f,\varphi _f\mid i\in I, f\in \overrightarrow{F}\}$ over the complete graph $\overrightarrow{ K(I)}=(I,\overrightarrow{F})$. In Sect. 2 we saw that amalgams of type ${\fancyscript{G}}$ are classified by pointings of an associated weak graph of groups $({\mathbb {A}},\overrightarrow{ K(I)})$. However, by Remark 2.6, for the purposes of classifying CT-structures of type ${\fancyscript{G}}$ we may discard all edges that do not belong to the Dynkin diagram and classify the amalgams whose type is the subamalgam of ${\fancyscript{G}}$ over the Dynkin diagram $\overrightarrow{\varGamma }$ instead.

In this section we shall take advantage of the special nature of CT-structures with simply laced diagram to define a graph of groups that is equally effective in classifying CT-structures, but whose vertex and edge groups are smaller than those of the weak graph of groups ${\mathbb {A}}$.

We shall only wish to classify those amalgams ${\fancyscript{G}}$ of type ${\fancyscript{G}}_0$ over $\varGamma$ that have property (D). Using Lemma 3.16 one finds that, for each $i\in I$ and each $e\in \overrightarrow{E}$,

Corollary 4.1

$$\begin{aligned} \mathrm{Aut}_{G_i}(D_i)&\cong T_i\rtimes (\langle \omega \rangle \times \mathrm{Aut}(k))\\ \mathrm{Aut}_{G_e}(\bar{G}_e,\bar{G}_{\bar{e}})&\cong T_{e,\bar{e}}\rtimes (\langle \omega \rangle \times \mathrm{Aut}(k)). \end{aligned}$$

where $T_i\unlhd \mathrm{Aut}_{G_i}(D_i)$ and $T_{e,\bar{e}}\unlhd \mathrm{Aut}_{G_e}(\bar{G}_e,\bar{G}_{\bar{e}})$ are the normal subgroups of diagonal automorphisms in the respective groups.

Note that the complements to $T_i$ and $T_{e,\bar{e}}$ are both isomorphic to ${\mathbb Z}_2\times \mathrm{Aut}(k)$, and are unique up to conjugation by elements in $T_i$ and $T_{e,\bar{e}}$ respectively.

Definition 4.2

Let ${\fancyscript{G}}$ be a CT-structure with simply-laced diagram $\varGamma$ that has property (D). As in Definition 3.13 ${\fancyscript{G}}$ determines a unique collection of tori ${\fancyscript{D}}({\fancyscript{G}})=\{D_i,D_f, D_{f,\bar{f}}\mid i\in I, f\in \overrightarrow{F}\}$. With the notation from Corollary 4.1, for $f\in \overrightarrow{F}-\overrightarrow{E}$, we let

$$\begin{aligned} T_f = \mathrm{ad}(\varphi _f^{-1})(T_{d_0(f)})&\qquad \text{ and }&T_{f,\bar{f}}=T_f\times T_{\bar{f}}\le \mathrm{Aut}_{G_f}(D_{f,\bar{f}}). \end{aligned}$$

From Corollary 4.1 it is clear that the set

$$\begin{aligned} {\fancyscript{T}}({\fancyscript{G}})&=\{T_i,T_f,T_{f,\bar{f}}\mid i\in I, f\in \overrightarrow{F}\}, \end{aligned}$$

is also uniquely determined by ${\fancyscript{G}}$.

Lemma 4.3

Given any collection $\{\tau _i\in T_i\mid i\in I\}$, there exist unique $\tau _{f,\bar{f}}\in T_{f,\bar{f}}$ such that $\tau =\{\tau _i,\tau _{f,\bar{f}}\mid i\in I, f\in \overrightarrow{F}\}$ induces an automorphism of ${\fancyscript{G}}$.

Proof

First we note that, for each $f\in \overrightarrow{F}$, $\bar{\tau }_f=\mathrm{ad}(\varphi _f^{-1})(\tau _{d_0(f)})$ is a diagonal (linear) automorphism of $\bar{G}_f\le G_f$. By definition, $\bar{\tau }_f\mathbin { \circ } \varphi _f=\tau _{d_0(f)}\mathbin { \circ } \varphi _f$.

If $f\in \overrightarrow{F}-\overrightarrow{E}$, then we define $\tau _{f,\bar{f}}$ to be the product $\bar{\tau }_f\times \bar{\tau }_{\bar{f}}:\bar{G}_f\times \bar{G}_{\bar{f}}\rightarrow \bar{G}_f\times \bar{G}_{\bar{f}}$. If $f\in \overrightarrow{E}$, suppose that with respect to some basis $\{v_1,v_2,v_3\}$ of eigenvectors for $D_f$ and $D_{\bar{f}}$ we have $\bar{\tau }_f=\mathrm{diag}\{a,b,1\}$ and $\bar{\tau }_{\bar{f}}=\mathrm{diag}\{1,c,d\}$, then let $\tau _{f,\bar{f}}=\mathrm{diag}\{ac,bc,bd\}$. In either case $\tau _{f,\bar{f}}\in \mathrm{Aut}_{G_f}(D_{f,\bar{f}})$. That $\tau$ commutes with the connecting maps follows easily.

Definition 4.4

(Basis for ${\fancyscript{G}}$) Let ${\fancyscript{G}}=\{ G_i,G_f,\varphi _f\mid i\in I, f\in \overrightarrow{F}\}$ be a CT structure with simply-laced diagram $\varGamma =(I,E)$ that has property (D). A basis of ${\fancyscript{G}}$ is a collection $\mathcal {V}=\{\mathcal {V}_e, \mathcal {V}_{\bar{e}}\mid e\in \overrightarrow{E}\}$ so that $\mathcal {V}_e=(v_1^e,v_2^e,v_3^e)$ is an (ordered) standard basis in $V_e$ for the standard pair $(\bar{G}_e, \bar{G}_{\bar{e}})$ of $G_e$ (see Definition 3.4) and $\mathcal {V}_{\bar{e}}=(v_1^{\bar{e}}=v^e_3,v_2^{\bar{e}}=v_2^e,v_3^{\bar{e}}=v_1^e)$ is the same basis but the ordering is reversed. Note that each $1$-space spanned by an element of $\mathcal {V}_e$ is stabilized by $D_e$ and $D_{\bar{e}}$. The edge reversal map is the element $\rho _e$ of $\mathrm{GL}(V_e)$ defined by $v_i^e\leftrightarrow v_i^{\bar{e}}$, for $i=1,2,3$.

Definition 4.5

(Concrete graph of groups) Let ${\fancyscript{G}}$ be a CT-structure with simply-laced Dynkin diagram $\varGamma =(I,E)$ that has property (D). Let $\mathcal {V}$ be a basis for ${\fancyscript{G}}$ as in Definition 4.4. For each $i\in I$ let ${\mathcal {W}}_i=(w_1^i,w_2^i)$ be a basis for $W_i$ identifying $G_i=\mathrm{SL}_2(k)$ with $\mathrm{SL}(W_i)$ such that $D_i$ is diagonal with respect to ${\mathcal {W}}_i$.

Let $\psi _e:G_{d_0(e)}\rightarrow \bar{G}_e\le G_e$ be the isomorphism induced by the linear map that takes the ordered basis $(w_1^{d_0(e)},w_2^{d_0(e)})$ to $(v_1^e,v_2^e)$. Explicitly, if $G_{d_0(e)}$ (resp. $G_e$) are identified with $\mathrm{SL}_2(k)$ (resp. $\mathrm{SL}_3(k)$) via the bases ${\mathcal {W}}_{d_0(e)}$ (resp. $\mathcal {V}_e$), then $\psi _e$ is given by

$$\begin{aligned} \psi _e:G_{d_0(e)}=\mathrm{SL}_2(k)&\rightarrow \mathrm{SL}_3(k)=G_e\\ A&\mapsto \begin{pmatrix} A &{} 0 \\ 0 &{} 1\end{pmatrix}. \end{aligned}$$

This defines a graph of groups $(\mathbb {K},\overrightarrow{\varGamma })$ in the following way. We let $\mathbb {K}=\{K_i,K_e,\psi _e\mid i\in I, e\in \overrightarrow{E}\}$ where $K_i$ is the complement in $\mathrm{Aut}_{G_i}(D_i)$ to $T_i$ defined with respect to ${\mathcal {W}}_i=(w_1^i,w_2^i)$ and $K_e$ is the complement in $\mathrm{Aut}_{G_e}(\bar{G}_e,\bar{G}_{\bar{e}})$ to $T_{e,\bar{e}}$, defined by $\mathcal {V}_e$ (See Lemma 3.16). Note that this yields $K_e=K_{\bar{e}}$, as desired. Finally, we may define the map $\kappa _e:K_e\rightarrow K_{d_0(e)}$ as given by the restriction of $\mathrm{ad}(\psi _e)$ to $K_e$, as the following lemma shows. We call $(\mathbb {K},\overrightarrow{\varGamma })$ the concrete graph of groups for ${\fancyscript{G}}$ associated with $\mathcal {V}$.

Remark 4.6

(on Definition 4.4) Note that by Lemma 3.5 each $\mathcal {V}_e$ is uniquely determined by the type of ${\fancyscript{G}}$ up to a linear transformation that is diagonal with respect to $\mathcal {V}_e$; it is independent of the connecting maps $\varphi _e$.

Remark 4.7

(on Definition 4.5):

• (a) Note that as in the proof of Lemma 3.5, since $k$ has at least four elements, ${\mathcal {W}}_i$ is uniquely determined up to a permutation and a diagonal linear transformation.

• (b) The choice of $\mathcal {V}$ and ${\mathcal {W}}$ does not affect the maps $\varphi _f$ ($f\in \overrightarrow{F}$); it only affects the way they are represented with respect to these bases. In particular, $\varphi _e^{-1}\psi _e\in \mathrm{Aut}_{G_{d_0(e)}}(D_{d_0(e)})=T_{d_0(e)}\rtimes K_{d_0(e)}$.

• (c) The concrete graph of groups $(\mathbb {K},\overrightarrow{\varGamma })$ in Definition 4.5 depends on ${\fancyscript{G}}$, $\mathcal {V}$ and ${\mathcal {W}}$, but is not related to the particular connecting maps $\varphi _f$ ($f\in \overrightarrow{E}$). Lemma 4.9 below shows that $\mathbb {K}$ is uniquely determined by the type of ${\fancyscript{G}}$ up to central isomorphism.

Remark 4.8

Note that if $({\mathbb {A}},\overrightarrow{\varGamma })$ is the weak graph of groups associated to ${\fancyscript{G}}$ as in Sect. 2, where we have removed all edges $f\in \overrightarrow{E}-\overrightarrow{F}$ using Lemma 2.5 then, for all $i\in I$ and $e\in \overrightarrow{E}$ we have

$$\begin{aligned} K_i\le A_i,&\qquad K_e\le A_e, \text{ and }&\kappa _e = \alpha _e|_{K_e}. \end{aligned}$$

Lemma 4.9

Let ${\fancyscript{G}}$ and ${\fancyscript{G}}'$ be CT-structures over $\varGamma$ with property (D) and let $(\mathbb {K},\overrightarrow{\varGamma })$ and $(\mathbb {K}',\overrightarrow{\varGamma })$ be the concrete graphs of groups associated as in Definition 4.5. If ${\fancyscript{G}}'$ is of type ${\fancyscript{G}}$, then there is a central isomorphism $(\mathbb {K},\overrightarrow{\varGamma })\rightarrow (\mathbb {K}',\overrightarrow{\varGamma })$ that only depends on a base change $(\mathcal {V},{\mathcal {W}})\mapsto (\mathcal {V}',{\mathcal {W}}')$.

Proof

First note that ${\fancyscript{G}}$ and ${\fancyscript{G}}'$ have the same groups $G_i$ and $G_e$ for $i\in I$ and $e\in \overrightarrow{E}$. Now the construction of $(\mathbb {K},\overrightarrow{\varGamma })$ does not depend on the connecting maps $\varphi _f$ and $\varphi _f'$, but only involves the maps $\psi _e$, which in turn depend uniquely on the basis $\mathcal {V}$ for ${\fancyscript{G}}$ and the collection ${\mathcal {W}}=\{{\mathcal {W}}_i=(w_1^i,w_2^i)\mid i\in I\}$ of bases chosen for the $W_i$. We now show that any other choice of $\mathcal {V}$ and ${\mathcal {W}}$ merely induces a central isomorphism between the resulting graphs of groups. Let $\mathcal {V}'$ and ${\mathcal {W}}'$ be another choice of a basis for ${\fancyscript{G}}$ and the $W_i$’s and let $(\mathbb {K}',\overrightarrow{\varGamma })$ be the resulting concrete graph of groups. For each $i\in I$, let $t_i\in \mathrm{Aut}(G_i)$ be induced by the linear map sending the ordered basis ${\mathcal {W}}_i$ to ${\mathcal {W}}_i'$ and for each $e\in \overrightarrow{E}$, let $t_e\in \mathrm{Aut}(G_e)$ be induced by the linear map sending the ordered basis $\mathcal {V}_e$ to $\mathcal {V}_e'$. Then the following diagram is commutative.

Since the bases defining the complements $K_e$, $K_e'$, $K_{d_0(e)}$ and $K_{d_0(e)}'$ all correspond via the maps in this diagram, also these complements themselves correspond to each other via the adjoint maps. This shows that the map $\phi :(\mathbb {K}',\overrightarrow{\varGamma })\rightarrow (\mathbb {K},\overrightarrow{\varGamma })$ given by $\phi =\{\phi _e=\mathrm{ad}(t_e):K_e'\rightarrow K_e,\phi _{i}=\mathrm{ad}(t_{i}):K_i'\rightarrow K_i\mid i\in I, e\in \overrightarrow{E}\}$ is an isomorphism which is central because all maps are given as conjugation by a (linear) base-change.

It is our aim to classify all CT structures of type ${\fancyscript{G}}_0$ with property (D) up to isomorphism. As a consequence of Remarks 4.6 and 4.7 and Lemma 4.9, we can then fix the following:

$$\begin{aligned} \overrightarrow{\varGamma }&=(I,\overrightarrow{E}),\\ {\fancyscript{G}}_0&=\{G_i,G_e,\psi _e\mid i\in I, e\in \overrightarrow{E}\},\\ {\fancyscript{D}}_0&=\{D_i, D_f,D_{f,\bar{f}}\mid i\in I, f\in \overrightarrow{F}\},\\ {\fancyscript{T}}_0&=\{T_i,T_f,T_{f,\bar{f}}\mid i\in I, f\in \overrightarrow{F}\},\\ \mathcal {V}_0&=\{\mathcal {V}_e,\mathcal {V}_{\bar{e}}\mid e\in \overrightarrow{E}\},\\ {\mathcal {W}}_0&= \{{\mathcal {W}}_i\mid i\in I\},\\ \mathbb {K}_0&= \{K_i,K_e,\kappa _e\mid i\in I, e\in \overrightarrow{E}\}. \end{aligned}$$

where ${\fancyscript{G}}_0$ is a Curtis-Tits structure having property (D) with simply-laced diagram $\varGamma$, ${\fancyscript{D}}_0$ is the collection of groups uniquely determined by ${\fancyscript{G}}_0$ as in Definition 3.13, ${\fancyscript{T}}$ is the collection of groups uniquely determined by ${\fancyscript{G}}_0$ as in Definition 4.2, $\mathcal {V}_0$ is a basis for ${\fancyscript{G}}_0$, and ${\mathcal {W}}_0$, $\psi _e$, and $\mathbb {K}_0$ are as defined in Definition 4.5.

We note that in order to describe all possible CT structures ${\fancyscript{G}}$ of type ${\fancyscript{G}}_0$ with property (D) up to isomorphism, in view of Remarks 2.6, 4.6 and 4.7, it suffices to make a choice for the maps $\{\varphi _e\mid e\in \overrightarrow{E}\}$ in such a way that $\varphi _e^{-1}\psi _e\in \mathrm{Aut}_{G_{d_0(e)}}(D_{d_0(e)})=T_{d_0(e)}\rtimes K_{d_0(e)}$, for all $e\in \overrightarrow{E}$. Our next aim is to show that modification of $\varphi _e$ by elements in $T_{d_0(e)}$ and $T_{e,\bar{e}}$ results in isomorphic amalgams and that these modifications can be used to ensure that $\varphi _e^{-1}\psi _e\in K_{d_0(e)}$.

Definition 4.10

Let ${\fancyscript{G}}$ be a CT-structure of type ${\fancyscript{G}}_0$ given by a family of connecting maps $\{\varphi _e\mid e\in \overrightarrow{E}\}$. We say that ${\fancyscript{G}}$ is a concrete CT structure associated to $(\mathcal {V},{\mathcal {W}})$ if its inclusion maps $\varphi _e$ satisfy $\mathrm{ad}(\varphi _e)(K_e)=K_{d_0(e)}$, where $K_e,K_{d_0(e)}\in \mathbb {K}_0$.

Note that, by definition, ${\fancyscript{G}}_0$ itself is a concrete CT-structure associated to $(\mathcal {V}_0,{\mathcal {W}}_0)$.

Lemma 4.11

Let ${\fancyscript{G}}'=\{G_i, G_f, \varphi _f' \mid i\in I,f\in \overrightarrow{F}\}$ be a CT structure of type ${\fancyscript{G}}_0$ that has property (D). Then, given a collection $\{K_i'\le \mathrm{Aut}_{G_i}(D_i)\mid i \in I\}$ of complements to the groups of diagonal automorphisms $T_i$, there exists a basis $\mathcal {V}'=\{\mathcal {V}_e',\mathcal {V}_{\bar{e}}'\mid e\in \overrightarrow{E}\}$ and a collection $\{K_e'\mid e\in \overrightarrow{E}\}$ of complements to $T_{e,\bar{e}}$ such that, for each $e\in \overrightarrow{E}$, $K_e'$ corresponds to $\mathcal {V}_e'$ and $\mathrm{ad}(\varphi _e')(K_e') = K_{d_0(e)}'$. Given $\{K_i'\mid i\in I\}$, the collection $\mathbb {K}'=\{K_i',K_e'\mid i\in I,e\in \overrightarrow{E}\}$ is unique and the bases $\mathcal {V}_e'$ are unique up to multiplication by a scalar in $\text{ Fix }(\mathrm{Aut}(k)).$

Proof

We shall use an automorphism $\tau =\{\tau _{d_0(e)},\tau _{e,\bar{e}}\mid f\in \overrightarrow{F}\}$ of ${\fancyscript{G}}_0$ as in Lemma 4.3 to establish the existence of $\mathcal {V}'$ and the collection $\mathbb {K}'$.

Recall that the groups $D_e$ and $T_e$ ($e\in \overrightarrow{E}$) are uniquely determined by the type of ${\fancyscript{G}}_0$ and are therefore shared by ${\fancyscript{G}}'$. The group $T_e$ acts—via $\mathrm{ad}(\cdot )$—transitively on the set of its complements, while acting on the corresponding bases. Two bases correspond to the same complement if and only if one is obtained from the other by permuting their elements and multiplying them by a scalar that is fixed by $\mathrm{Aut}(k)$. This proves the uniqueness part of the theorem.

For the existence we first pick a random base $\mathcal {V}''$ and modify it as follows. If $e\in \overrightarrow{E}$ then $\mathcal {V}_e''$ determines $K''_e$, a complement to $T_{e,\bar{e}}$. Restriction to $G_{d_0(e)}$ and $G_{d_0(\bar{e})}$ determines complements $K_{d_0(e)}'$ and $K_{d_0(\bar{e})}'$ to $T_{d_0(e)}$ and $T_{d_0(\bar{e})}$. These are conjugates of $K_{d_0(e)}$ and $K_{d_0(\bar{e})}$ under diagonal automorphisms $\tau _{d_0(e)}\in T_{d_0(e)}$ and $\tau _{d_0(\bar{e})}\in T_{d_0(\bar{e})}$. As in the proof of Lemma 4.3 there exists an automorphism $\tau =\{\tau _{d_0(e)},\tau _{e,\bar{e}}\mid f\in \overrightarrow{F}\}$ of ${\fancyscript{G}}_0$, where, for each $e\in \overrightarrow{E}$, $\tau _{e,\bar{e}}\in T_{e,\bar{e}}$ restricts to $\tau _{d_0(e)}$ and $\tau _{d_0(\bar{e})}$. The adjoint map $\mathrm{ad}(\tau _{e,\bar{e}})$ sends $K_e''$ to a complement $K_e'$ satisfying the statement of the lemma for the edge $e$, while the underlying linear map transforms the basis $\mathcal {V}''_e$ to the desired basis $\mathcal {V}_e'$.

Corollary 4.12

Let ${\fancyscript{G}}'=\{G_i, G_f, \varphi _f' \mid i\in I, f\in \overrightarrow{F}\}$ be a CT-structure of type ${\fancyscript{G}}_0$ that has property (D). Then, ${\fancyscript{G}}'$ is isomorphic to a concrete amalgam ${\fancyscript{G}}$ of type ${\fancyscript{G}}_0$. Moreover, the isomorphism can be taken to be diagonal.

Proof

By Lemma 4.11, given the collection $\{K_i\mid i\in I\}\subseteq \mathbb {K}_0$, there exists a basis $\mathcal {V}'$ and a collection $\mathbb {K}'=\{K_i',K_e'\mid i\in I,e\in \overrightarrow{E}\}$ that satisfies $K_i'=K_i$ for all $i\in I$, and $\mathrm{ad}(\varphi _e')(K_e') = K_i'$ for all $e\in \overrightarrow{E}$.

We now define an isomorphism $\phi =\{\phi _i,\phi _e\mid i\in I, e\in \overrightarrow{E}\}$ from a concrete amalgam ${\fancyscript{G}}$ to ${\fancyscript{G}}'$. Recall that $\mathcal {V}$ is the basis corresponding to the complements $K_e$. Now, for $e\in \overrightarrow{E}$, let $\phi _e:G_e\rightarrow G_e$ be the isomorphism induced by the (diagonal) linear map sending $\mathcal {V}_e$ to $\mathcal {V}'_e$ and let $\phi _i=\mathrm{id}_{G_i}$ for all $i\in I$. Now define ${\fancyscript{G}}$ by setting $\varphi _e=\phi _e^{-1}\mathbin { \circ } \varphi _e'\mathbin { \circ }\phi _{d_0(e)}$. Note that ${\fancyscript{G}} =\{G_i, G_e, \varphi _e \mid i\in I , e\in \overrightarrow{E}\}$ is concrete since $\mathrm{ad}(\varphi _e)(K_e)=\mathrm{ad}(\varphi _e')\mathbin { \circ } \mathrm{ad}({\phi _e}^{-1})(K_e)=K_{d_0(e)}$. Clearly $\phi$ defines an isomorphism between ${\fancyscript{G}}$ and ${\fancyscript{G}}'$.

As a consequence of Corollary 4.12, we only have to classify—up to isomorphism—those CT-structures of type ${\fancyscript{G}}_0$ with property (D) that are concrete, which means that we can assume that $\varphi _e^{-1}\psi _e\in K_{d_0(e)}\in \mathbb {K}_0\le {\mathbb {A}}_0$ for all $e\in \overrightarrow{E}$.

Theorem 4.13

For any simply-laced Dynkin diagram $\overrightarrow{\varGamma }$, there is a natural bijection between the set of isomorphism classes of concrete CT-structures of type ${\fancyscript{G}}_0$ with property (D) over $\overrightarrow{\varGamma }$ and the set of isomorphism classes of pointings of the concrete graph of groups $(\mathbb {K}_0,\overrightarrow{\varGamma })$.

Proof

Let ${\fancyscript{G}}$ be a concrete CT structure of type ${\fancyscript{G}}_0$ with property (D) over $\overrightarrow{\varGamma }$. Then ${\fancyscript{G}}$ defines a pointing of $\mathbb {K}_0$ by setting $\delta _e=\varphi _e^{-1}\mathbin { \circ }\psi _e$, for each $e\in \overrightarrow{E}$.

Conversely, given a pointing $((\mathbb {K}_0,\overrightarrow{\varGamma }),\delta )$ we define a CT-structure ${\fancyscript{G}}$ over $\overrightarrow{\varGamma }$ setting $\varphi _e=\psi _e\mathbin { \circ }\delta _e^{-1}$ for each $e\in \overrightarrow{E}$. In view of Remark 3.8 for $f\in \overrightarrow{F}-\overrightarrow{E}$ we can let $\varphi _f=\psi _f$.

The fact that the collection $\{\varphi _f\mid f\in \overrightarrow{F}\}$ defines a concrete CT-structure is immediate since, for $e\in \overrightarrow{E}$, $\mathrm{ad}(\varphi _e)=\mathrm{ad}(\delta _e^{-1})\mathbin { \circ }\mathrm{ad}(\psi _e)$, where $\mathrm{ad}(\psi _e)$ takes $K_e$ to $K_{d_0(e)}$ and $\mathrm{ad}(\delta _e^{-1})$ preserves $K_{d_0(e)}$.

It now suffices to show that the correspondence preserves isomorphism classes. In view of Remark 4.8, which says that $\mathbb {K}_0$ is obtained from ${\mathbb {A}}_0$ by selecting subgroups and restricting the connecting maps, and the observation that the correspondence described here is as in Definition 2.14, this follows from (the proof of) Theorem 2.15.

Corollary 4.14

For simply-laced Dynkin diagram $\overrightarrow{\varGamma }$, there is a natural bijection between the set of isomorphism classes of (not necessarily concrete) CT-structures of type ${\fancyscript{G}}_0$ with property (D) over $\overrightarrow{\varGamma }$ and the set of isomorphism classes of pointings of the concrete graph of groups $(\mathbb {K}_0,\overrightarrow{\varGamma })$.

Proof

This follows from Corollary 4.12 and Theorem 4.13.

Theorem 1.1 is now a consequence of Lemma 3.12, Corollary 4.14, Theorem 2.24 and Corollary 2.23.

5 The Curtis-Tits Theorem

5.1 Amalgams Resulting from the Curtis-Tits Theorem

Let $G$ be a simply connected Kac-Moody group that is locally split over a field $k$ (in the sense of [17], that is, the rank-$1$ and $2$ groups are split algebraic groups) with an admissible simply laced Dynkin diagram $\varGamma$ over some finite index set $I$. We shall prove that $G$ possesses an amalgam that is an orientable Curtis-Tits structure. Let $(W,\{r_i\}_{i\in I})$ be the Coxeter system of type $\varGamma$. Then $G$ has a twin BN-pair $(B^+,N,B^-)$ of type $\varGamma$, which gives rise to a Moufang twin-building $\varDelta =(\varDelta _+,\varDelta _-,\delta _+,\delta _-,\delta _*)$ of type $\varGamma$, where, for $\varepsilon =\pm$ we have $\varDelta _\varepsilon =G/B^\varepsilon$ and

$$\begin{aligned} \begin{array}{rll} \delta _\varepsilon (gB^\varepsilon ,hB^\varepsilon )= w \in W &{}&{} \qquad \quad \text{ whenever } B^\varepsilon g^{-1}hB^\varepsilon =B^\varepsilon w B^\varepsilon ,\\ \delta _*(gB^+,hB^-)= w\in W &{}&{} \qquad \quad \text{ whenever } {B^+g^{-1}hB^-=B^+wB^-}. \end{array} \end{aligned}$$

Two chambers $C$ and $D$ are called opposite if $\delta _*(C,D)=1$. Fix two opposite chambers $C^+=B^+$ and $C^-=B^-$.

Since $\varGamma$ is simply laced, by [18] the local structure of $\varDelta$ determines the global structure. The generalization of the Curtis-Tits theorem to $2$-spherical diagrams given in [1] yields $G$ as the universal completion of an amalgam of rank $1$ and rank $2$ Levi complements.

Since $\varGamma$ is simply-laced, by [12], $G$ is a central quotient of the universal completion of the following amalgam:

$$\begin{aligned} {\fancyscript{R}}&=\{R_i,R_f, \rho _f\mid i\in I,f\in \overrightarrow{F}\}, \end{aligned}$$

where $R_i=\langle U_i^+,U_i^-\rangle$, $R_f=\langle R_{d_0(f)},R_{d_0(\bar{f})}\rangle$, and $\{U_i^+\mid i\in I\}$ is a selection of positive root groups corresponding to a fundamental system of positive roots, and $\rho _f:R_{d_0(f)}\hookrightarrow R_f$ is given by inclusion of subgroups in $G$.

Remark 5.1

Note that an automorphism of $\varGamma$ corresponds to a renaming of the types of $\varDelta$, and a corresponding renaming of the groups in the amalgam ${\fancyscript{R}}$. Naturally the buildings corresponding to these respectively labeled amalgams are isomorphic via a type permuting isomorphism. In accordance with our convention for amalgams in Definition 2.1 and Remark 2.3 we shall regard such buildings and amalgams as being different. Note that Tits uses the same convention in [28].

The fact that $G$ is locally split means that whenever $e\in \overrightarrow{E}$, then the $e$-residue on $C^+$ is isomorphic to the building associated to the group $\mathrm{SL}_3(k)$. A priori it is possible that the actual groups $R_i$ and $R_f$ are proper quotients of $\mathrm{SL}_2(k)$ and $\mathrm{SL}_3(k)$. However, the following is true.

Lemma 5.2

Assume that the Dynkin diagram $\varGamma$ is simply laced and admissible and that $|I|\ge 4$. Then, for all $i\in I$ and $f\in \overrightarrow{F}$,

$$\begin{aligned} R_i&\cong \mathrm{SL}_2(k)\\ R_f&\cong {\left\{ \begin{array}{ll} \mathrm{SL}_3(k) &{} \text{ if } f\in \overrightarrow{E}\\ \mathrm{SL}_2(k)\circ \mathrm{SL}_2(k) &{} \text{ else. } \end{array}\right. } \end{aligned}$$

This follows from the general result in [12] and the observation that on any subdiagram of type $A_3$, the rank $1$ and $2$ groups intersect the center of the corresponding rank $3$ group trivially.

The fact that $G$ has a torus generated by a coherent selection of tori in each $R_i$ ($i\in I$) implies that ${\fancyscript{R}}$ has property (D). The fact that ${\fancyscript{R}}$ is oriented follows from the observation that for each $i$, the root group $U^+_i$ of the fundamental positive root ${\alpha _i}$ belongs to $R_i$ and $U^+_i\subseteq B^+$. In particular, for any $f\in \overrightarrow{F}$, $U^+_{d_0(f)}$ and $U^+_{d_0(\bar{f})}$ belong to a common Borel group of $R_f$. In sum, ${\fancyscript{R}}$ is an oriented Curtis-Tits structure (in particular it has property (D)) with Dynkin diagram $\varGamma$ for $G$.

In the remainder of this section, we shall prove that every oriented CT-structure with admissible Dynkin diagram can be obtained as the Curtis-Tits amalgam of some simply connected Kac-Moody group that is locally split over $k$.

Let $\varGamma$ be an admissible Dynkin diagram and ${\fancyscript{G}}(\varGamma )$ an oriented CT-structure over some field $k$. The fact that ${\fancyscript{G}}(\varGamma )$ is oriented allows us to define a Moufang foundation, which by a result of Mühlherr is integrable to a twin-building $\varDelta$.

5.2 Moufang Foundations and Orientable CT-Amalgams

We shall make use of the following definition of a foundation [17], which is equivalent to the definition in [27]:

Definition 5.3

Let $\varGamma$ be an admissible Dynkin diagram over $I$. A foundation of type $\varGamma$ is a triple

$$\begin{aligned} (\{\varDelta _e\mid e\in \overrightarrow{E}\},\{C_e\mid e\in \overrightarrow{E}\},\{\theta _e^f\mid e,f\in \overrightarrow{E} \text{ with } d_0(e)=d_0(f)\}), \end{aligned}$$

satisfying the following conditions:

• (Fo1) $\varDelta _e=\varDelta _{\bar{e}}$ is a building of type $A_2$ for each $e\in \overrightarrow{E}$;

• (Fo2) $C_e=C_{\bar{e}}$ is a chamber of $\varDelta _e$ for each $e\in \overrightarrow{E}$;

• (Fo3) $\theta _e^f$ is a bijection between the ${d_0(e)}$-panel on $C_e$ in $\varDelta _e$ and the $d_0(f)$-panel on $C_f$ in $\varDelta _f$ such that $\theta _e^f(C_e)=C_f$ and if $e,f,g\in \overrightarrow{E}$ are such that $d_0(e)=d_0(f)=d_0(g)$, then $\theta _f^g\mathbin { \circ }\theta _e^f=\theta _e^g$.

This foundation is said to be of Moufang type if $\varDelta _e$ is a Moufang building for each $e\in \overrightarrow{E}$ and if in (Fo3) the map $\theta _e^f$ induces an isomorphism between the Moufang set induced by $\varDelta _e$ on the $d_0(e)$-panel of $C_e$ and the Moufang set induced by $\varDelta _f$ on the $d_0(f)$-panel of $C_f$.

We shall now describe how to obtain a Moufang foundation from a given orientable CT-structure $({\fancyscript{G}},\overrightarrow{\varGamma })$. Let $\{X^+_i,\mid i\in I\}$ be the collection of root groups as in Definition 3.14 and let $\{B^+_{e}\mid e\in \overrightarrow{E}\}$ be the collection of Borel groups in $G_{e}$ such that $\varphi _{e}(X^+_{d_0(e)})$ and $\varphi _{\bar{e}}(X^+_{d_0(\bar{e})})$ are contained in $B^+_{e}$ for any $e\in \overrightarrow{E}$ (note that this in fact determines $B^+_{e}$ uniquely). For each $e\in \overrightarrow{E}$, let $\varDelta _{e}$ be the Moufang building of type $A_2$ obtained from $G_{e}$ via the BN-pair $(B^+_{e},N_{G_{e}}(D_{e}))$ and let $C_{e}$ be the chamber given by $B^+_{e}$. Now let $e,f\in \overrightarrow{E}$ be such that $d_0(e)=d_0(f)$. Let $n_e$ be the element of $N_{G_{e}}(D_{e,\bar{e}})$ given by

$$\begin{aligned} \begin{pmatrix} 0 &{} -1 &{} 0 \\ 1 &{} 0 &{} 0 \\ 0 &{} 0 &{} 1\\ \end{pmatrix}\end{aligned}$$

with respect to the ordered basis $\mathcal {V}_{e}$. Now, for every $e\in \overrightarrow{E}$, let ${\bar{X}_e^+}=\varphi _e({X_{d_0(e)}^+})$. Then, the $d_0(e)$-panel of $\varDelta _{e}$ containing $C_{e}$ equals

$$\begin{aligned} {{\bar{\varDelta }}_e}=\{C_e\}\cup \{\lambda n_e C_e\mid \lambda \in {\bar{X}_e^+}\}. \end{aligned}$$

We now define

$$\begin{aligned} \theta _e^f:\bar{\varDelta }_e&\rightarrow \bar{\varDelta }_f\\ C_e&\mapsto C_f\\ \lambda n_e C_e&\mapsto \varphi _f\mathbin { \circ }\varphi _e^{-1}(\lambda ) n_f C_f \end{aligned}$$

Note that since the structure of the $d_0(e)$-panel of $\varDelta _{e}$ on $C_{e}$ (resp. of $\varDelta _f$ on $C_f$) as a Moufang set is entirely determined by $G_{d_0(e)}$ the map $\theta _e^f$, induced by the group isomorphism $\varphi _f\mathbin { \circ }\varphi _e^{-1}$ preserves this structure. Clearly, if $g\in \overrightarrow{E}$ is such that $d_0(e)=d_0(f)=d_0(g)$, then $\theta _f^g\mathbin { \circ }\theta _e^f=\theta _e^g$. To prove soundness, we observe that since $\varGamma$ is admissible, the connected subdiagrams with three vertices are of type $A_3$ and the corresponding amalgam is the unique Curtis-Tits amalgam for $\mathrm{SL}_4(k)$.

By Lemma 3.15, for each $e\in \overrightarrow{E}$, the selection $\{X^\pm _{d_0(e)},X^\pm _{d_0(\bar{e})}\}$ determines a pair of opposite chambers, hence an apartment $\varSigma _e$ in $\varDelta _e$. Moreover, $\varSigma _e$ intersects $\bar{\varDelta }_e$ in the panel $\{C_e,n_e C_e\}$. Clearly then we have $\theta _e^f(n_eC_e)=n_f C_f$. This proves that the system ${\fancyscript{S}}=\{\varSigma _e=\varSigma _{\bar{e}}\mid e\in \overrightarrow{E}\}$ is an apartment of $\mathbf{F}$.

This proves the following.

Lemma 5.4

The triple

$$\begin{aligned} \mathbf{F}=(\{\varDelta _{e}\mid e\in \overrightarrow{E}\},\{C_{e}\mid e\in \overrightarrow{E}\},\{\theta _e^f\mid e,f\in \overrightarrow{E} \text{ with } d_0(e)=d_0(f)\}) \end{aligned}$$

obtained from the CT-structure $({\fancyscript{G}},\overrightarrow{\varGamma })$ as above, is a sound Moufang foundation. Moreover, the selection $\{X_i^+,X_i^-\mid i\in I\}$ determines a unique apartment in $\mathbf{F}$.

Proof

of Corollary 1.2. As proved in the beginning of Sect. 5, the amalgam ${\fancyscript{R}}$ arising from [12] is an orientable CT-structure with diagram $\overrightarrow{\varGamma }$.

Conversely, let ${\fancyscript{G}}$ be a concrete orientable CT structure with diagram $\overrightarrow{\varGamma }$ and select a system of root groups ${\fancyscript{X}}=\{X^\pm _i\mid i\in I\}$ and let $\mathbf{F}$ be the Moufang foundation with apartment ${\fancyscript{S}}$ constructed from ${\fancyscript{G}}$ as in Lemma 5.4. By the main result of [17] this sound Moufang foundation is integrable to a twin-building $\varDelta$ which is unique up to isomorphism. Let $(G, B^+,B^-,N)$ be the twin-BN-pair for the universal Kac-Moody group $G$ giving rise to $\varDelta$.

Identify $\mathbf{F}$ with its image $E_2(C^+)$ in $\varDelta$ where $C^+$ is the chamber represented by $B^+$. Then ${\fancyscript{S}}$ determines a chamber $C^-$ opposite to $C^+$ (see e.g. [17, Sect. 8]). By transitivity of $G$ on pairs of opposite chambers, we may assume $C^-$ corresponds to $B^-$. Pick any $e\in \overrightarrow{E}$. Then, $\bar{\varDelta }_e$ is identified with the $d_0(e)$-panel on $C^+$. Let $\varSigma =\varSigma (C^+,C^-)$ be the twin-apartment that is the coconvex hull of $C^+$ and $C^-$. Then, $\varSigma _e=\varSigma \cap \varDelta _e$. Let $U_{d_0(e)}^\pm$ be the root groups in $G$ stabilizing the roots of $\varSigma$ determined by the panel $\bar{\varDelta }_e$ on $C^+$. By [17, Theorem 8.1], we can extend the action of $X_{d_0(e)}^\pm$ on $\varDelta _e$ to $\varDelta$. This defines an isomorphism $X_{d_0(e)}^\pm \rightarrow U_{d_0(e)}^\pm$ that extends to a homomorphism $\phi _e:G_e\rightarrow R_e=\langle U_{d_0(e)}^\pm , U_{d_0(\bar{e})}^\pm \rangle _G$. From Lemma 5.2 it follows that this homomorphism is in fact an isomorphism yielding a commutative diagram:

6 Twists of Split Kac-Moody Groups

Let $G_\varGamma (k)$ be the split Kac-Moody group with Dynkin diagram $\varGamma$ over $k$ and let ${\fancyscript{G}}_0=\{G_i,G_f, \psi _f\mid i\in I, f\in \overrightarrow{F}\}$ be the CT-structure associated to $G_\varGamma (k)$ as in Sect. 5.1. We may assume that this amalgam is concrete by Corollary 4.12 and let $(\mathbb {K}_0,\overrightarrow{\varGamma })$ be the associated concrete graph of groups. This suggests the following definition.

Definition 6.1

For any pointing $((\mathbb {K}_0, \overrightarrow{\varGamma }), \delta )$ of the concrete graph of groups $(\mathbb {K}_0,\overrightarrow{\varGamma })$, the $\delta$ -twist is the universal completion $G_\varGamma ^\delta (k)$ of the Curtis-Tits amalgam corresponding to $\delta$ as in Theorem 4.13. More precisely, for any $e\in \overrightarrow{E}$ and $i \in I$ we get a copy $G_i=\mathrm{SL}_2(k)$ and $G_e=G_{\bar{e}}=\mathrm{SL}_3(k)$ and the relations given by those in $G_i$ and $G_e$ together with the following:

• (i) if $e\in \overrightarrow{E}$ then $\varphi _e=\psi _e\mathbin { \circ } \delta _e^{-1}:G_{d_0(e)}\hookrightarrow G_e$ identifies $G_{d_0(e)}$ with a subgroup of $G_e$;

• (ii) if $f\in \overrightarrow{F}-\overrightarrow{E}$, then $[G_{d_0(f)}, G_{d_0(\bar{f})}]=1$.

Two twists are considered equivalent if the corresponding amalgams, and hence pointings, are isomorphic. As for amalgams and buildings, two twists that differ by a non-trivial graph automorphism are considered non-equivalent.

We can now make an efficient selection of twists $\delta$. To that end, let us fix a spanning tree $\overrightarrow{T}$ of $\varGamma$, together with a set of directed edges $\overrightarrow{H}$ that does not intersect $\bar{\overrightarrow{H}}=\{\bar{e}\mid e\in \overrightarrow{H}\}$ and $\overrightarrow{F} =\overrightarrow{T} \cup \overrightarrow{H}\cup \bar{\overrightarrow{H}}$. We will construct an amalgam as follows. For each $e\in \overrightarrow{H}$ we take $\delta _e\in K_{d_0(e)}$. Now let

$$\begin{aligned} \varphi _e={\left\{ \begin{array}{ll} \psi _e\mathbin { \circ } \delta _{e}^{-1}&{} \text{ if } e\in \overrightarrow{H},\\ \psi _e&{} \text{ else. } \end{array}\right. } \end{aligned}$$

The resulting amalgam is denoted by ${\fancyscript{G}}^\delta$.

Corollary 6.2

Let $\varGamma$ be a connected simply laced Dynkin diagram with no triangles and $k$ a field with at least four elements and let $G$ be a universal Kac-Moody group with diagram $\varGamma$ that is locally split over $k$. Then, the amalgam arising from the Curtis-Tits theorem for $G$ is isomorphic to a unique Curtis-Tits amalgam ${\fancyscript{G}}^\delta$. Moreover, in that case $G$ is a central quotient of the universal completion of ${\fancyscript{G}}^\delta$.

Proof

Since the set $\overrightarrow{H}$ corresponds to a unique set of generators for the fundamental group of $\varGamma$, there is a natural bijection between sets $\{\delta _e\mid e\in \overrightarrow{H}\}$ and homomorphisms $\varPhi :\pi (\overrightarrow{\varGamma },i_0)\rightarrow \mathrm{Aut}(k)$. The result now follows from Corollary 1.2 and the main result from [12].

Remark 6.3

Note that Corollary 6.2 does not say that different twists necessarily correspond to non-isomorphic Kac-Moody groups. In fact this is false. For instance, if in Corrollary 6.2 one changes the labeling of $\varGamma$ via a graph automorphism, the resulting twists and Curtis-Tits amalgams will be considered to be non-equivalent, even though the corresponding groups and buildings are isomorphic via a type permuting isomorphism.

We would like to thank a careful referee for drawing our attention to this fact.

Corollary 1.3 follows immediately from Corollary 6.2.