Regular Algebraic Surfaces, Ramification Structures and Projective Planes

Abstract

Regular algebraic surfaces isogenous to a higher product of curves can be obtained from finite groups with ramification structures. We find unmixed ramification structures for finite groups constructed as $p$-quotients of particular infinite groups with special presentation related to finite projective planes.

Appendices

Appendix 1: Expanders Associated to the Group $G_0$

Expander graphs are defined with the help of the edge expansion rate.

Definition A. 1

Let $\mathcal G=(V,E)$ be a combinatorial graph with vertex set $V$ and edge set $E$. Then the edge expansion rate $h({\mathcal G})$ is defined as

$$ h({\mathcal G}) = \inf _{\mathrm{finite} A \subset V} \frac{|\partial A|}{\min (|A|,|V \backslash A|)}, $$

where $\partial A \subset E$ is the set of all edges connecting a vertex of $A$ with a vertex of $V \backslash A$.

Expanders are infinite families of finite graphs which are both sparse and highly connected. They are not only theoretically important but have also applications in computer science for, e.g., robust network designs.

Definition A. 2

A sequence ${\mathcal G}_n = (V_n,E_n)$ of connected finite graphs with $|V_n| \rightarrow \infty$ is called a family of expanders if there exists $k \ge 2$ and $\epsilon > 0$ such that

• (a) all graphs ${\mathcal G}_n$ are $k$-regular,

• (b) $h({\mathcal G}_n) \ge \epsilon$ for all $n$.

It was observed in [22] that the subgroup $H_0$ of $G_0$ generated by $x_0,x_1$ has index $2$, and that both groups $H_0$ and $G_0$ are just infinite and have Kazhdan property (T). Property (T) implies that, for a fixed choice of generators, the Cayley graphs of all quotients by finite index normal subgroups have a uniform positive lower bound for their edge expansion rate (see [19, Proposition 3.3.1]). A presentation of the subgroup $H_0$ is given by

$$ H_0 = \langle x_0,x_1 \mid r_1, r_2, r_3 \rangle , $$

where

$$\begin{aligned} r_1= & {} (x_1 x_0)^3 x_1^{-3} x_0^{-3}, \\ r_2= & {} x_1 x_0^{-1} x_1^{-1} x_0^{-3} x_1^2 x_0^{-1} x_1 x_0 x_1, \\ r_3= & {} x_1^3 x_0^{-1} x_1 x_0 x_1 x_0^2 x_1^2 x_0 x_1 x_0. \end{aligned}$$

We have the following Cayley graph expanders obtained from finite groups with just two generators and four relations.

Theorem A. 3

(cf. [22, Theorem 1]) The groups

$$ H_k = \langle x_0, x_1 \mid r_1, r_2, r_3, [x_1, \underbrace{x_0,\dots ,x_0}_{k}] \rangle $$

are finite with $|H_k| \rightarrow \infty$, and the associated Cayley graphs with respect to the generators $x_0,x_1$ define an infinite family of expanders of vertex degree 4.

Using the faithful matrix representation of $H_0$ by infinite upper triangular matrices and their truncations at the $k$th upper diagonal as mentioned in Sect. 3.1, we obtain another family $\widetilde{H}_k$ of finite nilpotent groups whose associated Cayley graphs ${\mathcal G}_k$ with respect to the generators $x_0,x_1$ are another family of expander graphs which form a tower of coverings

$$ \cdots {\mathcal G}_k \rightarrow {\mathcal G}_{k-1} \rightarrow \cdots \rightarrow {\mathcal G}_1 \rightarrow {\mathcal G}_0, $$

whose covering indices are powers of $2$ (for more details, see [22]). It was conjectured in [22, Conjecture 2] that the covering indices follow the pattern $4,8,4,8,8,4,8,8, 4,8,8,\dots$. See Fig. 2 for the graph ${\mathcal G}_2$. We use the notation $z_1 = [x_0,x_1], z_2 = x_0^2, z_3 = x_1^2$ and $z_{ij} = z_i z_j$ and $z_{ijk} = z_i z_j z_k$. The elements expressed by $z_i$ lie in the centre of $\widetilde{H}_2$. The same graph was illustrated in [22, Fig. 4], but the illustration given here is more symmetric. Solid edges from vertices with label $i$ to vertices with label $i+1$ (mod $4$) represent right multiplication by $x_0$, while dashed edges from vertices with label $i$ to vertices with label $i+1$ (mod $4$) represent right multiplication by $x_1$. Note that the solid $4$-cycles as well as the dashed $4$-cycles in ${\mathcal G}_3$ are consequences of $x_0^4 = x_1^4 = 1$ in $\widetilde{H}_2$.

Fig. 2

The graph ${\mathcal G}_2$

Another construction of $3$-regular expanders was given in [17]. Starting from the same groups $\widetilde{H}_k$, we now consider the associated Cayley graphs $X_k$ with respect to the generators $x_0,x_1,x_3$ where $x_3 = x_1^{-1} x_0^{-1}$. The graphs $X_k$ are $6$-regular and ${\mathcal G}_k$ is a subgraph of $X_k$ with the same number of vertices. Property (T) guarantees that the graphs $X_k$ are also a family of expanders. One can check that $X_k$ forms a tessellation of a closed Riemann surface by triangles and by $2^l$-gons (with $l$ only depending on $k$) and that every edge of $X_k$ belongs to precisely one triangle of $X_k$. Now we apply a $\Delta -Y$ transformation to the graphs $X_k$ to obtain new graphs $T_k$. The $\Delta -Y$ transformation removes the edges of every triangle in the original graph $X_k$, adds a new vertex in its centre, and connects this new central vertex with new edges to the $3$ original vertices of the triangle. It turns out that the vertex set of the new graph $T_k$ is twice as large as the vertex set of old graph $X_k$, and that $T_k$ is $3$-regular. Moreover, there is an explicit connection between the eigenfunctions of the adjacency matrix of $X_k$ and the eigenfunctions of the adjacency matrix of $T_k$ (see [17, Theorem 2.1]). The spectral characterisation of expander graphs then implies that the new family $T_k$ of $3$-regular graphs is, again, a family of expanders.

For yet another expander graphs construction from the group $G_0$ see [20, 23].

Appendix 2: Representation for the Group G

We include a representation for the group $G$ (given by (2)) in $\mathrm{GL}(9,\mathbb {F}_{3}[1/Y])$, which may be useful in the future (as the matrix representations for the group $G_{0}$ with presentation (1) were useful for several works [8, 20, 22]). The representation is due to Donald Cartwright and the algebra program REDUCE. Recall that the group $G$ coincides with the group $1.1$ in [9], where we relate the generators by $a_{i}=x_{2i}$ for $i=0,\ldots ,12$, with indices taken modulo $13$. We set

$$ x_{0}: \left( \begin{matrix} 1 &{} 1 &{} 1 &{} 0 &{} 2&{} 2&{}0 &{}1 &{}1 \\ 0 &{} 1 &{} 2 &{} 0 &{} 0&{} 1&{}0 &{}0 &{}2 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0&{} 0&{}0 &{}0 &{}0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 1&{} 1&{}0 &{}2 &{}2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1&{} 2&{}0 &{}0 &{}1 \\ 0&{} 0 &{} 0 &{} 0 &{} 0&{} 1&{}0 &{}0 &{}0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}1 &{}1 &{}1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0 &{}1 &{}2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0 &{}0&{}1 \\ \end{matrix}\right) +\frac{1}{Y} \left( \begin{matrix} 0 &{} 2 &{} 2 &{} 0 &{} 1&{} 1&{}0 &{}2 &{}2 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0&{} 2&{}0 &{}0 &{}1\\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0&{}0 &{}0 \\ 0 &{} 1 &{} 1 &{} 0 &{} 2&{} 2&{}0 &{}1 &{}1 \\ 0 &{} 0 &{} 2 &{} 0 &{} 0&{} 1&{}0 &{}0 &{}2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0 &{}0 &{}0 \\ 0 &{} 2 &{} 2 &{} 0 &{} 1&{} 1&{}0 &{}2 &{}2 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0&{} 2&{}0 &{}0 &{}1 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0 &{}0 &{}0 \\ \end{matrix}\right) $$

and

$$ \tau : \left( \begin{matrix} 1 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0 &{}0 &{}0 \\ 0 &{} 1 &{} 0 &{} 0 &{} 0&{} 0&{}0 &{}0 &{}0 \\ 0 &{} 0 &{} 1 &{} 0 &{} 0&{} 0&{}0 &{}0 &{}0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1&{} 0&{}0 &{}0 &{}0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 1&{} 1&{}0 &{}0 &{}0 \\ 0 &{} 0 &{} 0 &{} 1 &{} 0&{} 1&{}0 &{}0 &{}0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}2 &{}2 &{}0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}0 &{}1 &{}2 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0&{} 0&{}2 &{}0 &{}1 \\ \end{matrix}\right) , $$

where the other generators $x_{1},\ldots ,x_{12}$ are formed via conjugation of $x_{0}$ by $\tau$, i.e. $x_{i}=\tau ^{i}x_{0}\tau ^{-i}$ for $i=1,\ldots ,12$.

The idea in creating this representation is to write $\mathbb {F}_{27}=\mathbb {F}_{3}(\theta )$, where $\theta$ is a primitive element on $\mathbb {F}_{27}$ satisfying $\theta ^{3}=\theta +1$, and to use the basis $\{\theta ^{i}\sigma ^{j}|i,j=0,1,2\}$ for the divison algebra $\mathcal {A}$ over $\mathbb {F}_{27}(Y)$ for an indeterminate $Y$ (in the order $1,\theta , \theta ^{2}, \sigma , \theta \sigma , \ldots ,\theta ^2\sigma ^2$). Here $\sigma$ is assumed to satisfy $\sigma ^3 = Y - 1$ (which implies $(1+\sigma )^{-1}=(1/Y)(1-\sigma +\sigma ^{2})$) and $\sigma \theta \sigma ^{-1} = \theta ^3$. The generators of $\mathcal {T}_{\mathcal {K}}$, where $\mathcal {K}$ is a triangle presentation from [8, 9], are the $a_{u}=u^{-1}(1+\sigma )u$, where $u\in \mathbb {F}_{27}^{\times }/\mathbb {F}_3^{\times }$. Since $\mathbb {F}_{27}^{\times } = \mathbb {F}_3^{\times }\cdot \{ 1=\theta ^{13},\theta ,\dots ,\theta ^{12} \}$, we choose $\alpha _k=\theta ^{-k}(1+\sigma )\theta ^k$ as in [9, p. 178]. The $\alpha _k$’s act on $\mathcal {A}$ by conjugation. A straightforward calculation yields

$$\begin{aligned}&\alpha _k \theta ^i\sigma ^j \alpha _k^{-1} = \theta ^i \sigma ^j \frac{1}{Y} + \left( \theta ^{3i+2k} - \theta ^{i+2 \cdot 3^j k} \right) \sigma ^{j+1} \frac{1}{Y} \\&\qquad \qquad \quad + \left( \theta ^{i+8 \cdot 3^j k} - \theta ^{3i+2k+2\cdot 3^{j+1} k} \right) \sigma ^{j+2} \frac{1}{Y} + \theta ^{3i+2k+8 \cdot 3^{j+1}k} \sigma ^j \frac{Y-1}{Y}. \end{aligned}$$

Expressing the conjugation by $\alpha _k$ with respect to the above basis of $\mathcal {A}$ then gives rise to a representation as a $9 \times 9$ matrix over the field $\mathbb {F}_{3}(1/Y)$. We conclude from [9] that the matrices associated to the $\alpha _k$ satisfy the relations of our generators $x_k$. Note, finally, that the above matrix for $\tau$ represents the conjugation by $\theta$ in $\mathcal {A}$, i.e., $z \mapsto \theta ^{-1} z \theta$.