# Non-residually finite extensions of arithmetic groups

## Abstract

*G*is a simple algebraic group over the rational numbers satisfying both strong approximation, and the congruence subgroup problem. We show that every arithmetic subgroup of

*G*has finite extensions which are not residually finite. More precisely, we investigate the group

*G*. Elements of \({\bar{H}}^2(\mathbb {Z}/n)\) correspond to (equivalence classes of) central extensions of arithmetic groups by \(\mathbb {Z}/n\); non-zero elements of \({\bar{H}}^2(\mathbb {Z}/n)\) correspond to extensions which are not residually finite. We prove that \({\bar{H}}^2(\mathbb {Z}/n)\) contains infinitely many elements of order

*n*, some of which are invariant for the action of the arithmetic completion \({\widehat{G(\mathbb {Q})}}\) of \(G(\mathbb {Q})\). We also investigate which of these (equivalence classes of) extensions lift to characteristic zero, by determining the invariant elements in the group

*c*. When \(G(\mathbb {R})\) has no simple components of complex type, we prove that \(c=b+m\), where

*b*is the number of simple components of \(G(\mathbb {R})\) and

*m*is the dimension of the centre of a maximal compact subgroup of \(G(\mathbb {R})\). In all other cases, we prove upper and lower bounds on

*c*; our lower bound (which we believe is the correct number) is \(b+m\).

## Keywords

Cohomology of arithmetic groups Congruence subgroup property Residually finite group## Mathematics Subject Classification

11F77 11F06## 1 Introduction

An abstract group *G* is said to be residually finite if, for every non-trivial element *g*, there is a subgroup *H* of finite index in the group, which does not contain *g*. The content of this statement is not changed if we insist that *H* is a normal subgroup of *G*. This is equivalent to the statement that the canonical map from the group to its profinite completion is injective.

In this note, we show that a weaker version of Deligne’s result holds for a large class of arithmetic groups.

*G*is an algebraic group over \(\mathbb {Q}\), which is simple and simply connected. As we have seen above, the group \(G(\mathbb {R})\) may fail to be simply connected with the archimedean topology; this happens whenever a maximal compact subgroup of \(G(\mathbb {R})\) has infinite centre. We shall assume that the fundamental group \(\pi _{1}(G(\mathbb {R}))\) has more than 2 elements. We can define just as before an extension

*G*has the congruence subgroup property, then every subgroup of finite index in \({\tilde{\varGamma }}\) contains \(\ker \big (\pi _{1}(G(\mathbb {R})) \rightarrow \mu _{2}\big )\).

To show that this generalization is not vacuous, we remark that \(\pi _{1}(G(\mathbb {R}))\) is infinite whenever there is a Shimura variety associated to *G*, and the congruence subgroup property is known to hold for simple, simply connected groups of rational rank at least 2.

In this paper, we shall deal also with groups *G*, for which Deligne’s construction cannot be used. The most easily stated consequence of our results is the following.

### Theorem 1

*G*be a simple algebraic group over \(\mathbb {Q}\), which is algebraically simply connected, and has positive real rank. Assume also that

*G*and has finite congruence kernel. Let \(\varGamma \) be an arithmetic subgroup of \(G(\mathbb {Q})\). Then there is a finite abelian group

*A*and an extension of groups

### 1.1 Residual finiteness of hyperbolic groups

It is an important open question in geometric group theory whether every Gromov-hyperbolic group is residually finite (see for example [1, 9, 11, 15, 26]). This question turns out to be related to the following conjecture of Serre [20].

### Conjecture 1

Let \(G/\mathbb {Q}\) be a simple, simply connected algebraic group of real rank 1. Then the congruence kernel of *G* is infinite.

As a consequence of Theorem 1, we obtain the following.

### Corollary 1

If every Gromov–hyperbolic group is residually finite then Conjecture 1 is true.

### Proof

Let \(\varGamma \) be an arithmetic subgroup of a Lie group with real rank 1. It is known that \(\varGamma \) is Gromov–hyperbolic (see chapter 7 of [10]). Since hyperbolicity is invariant under quasi-isometry, every finite extension of \(\varGamma \) is also hyperbolic, and hence by assumption residually finite. If the congruence kernel were finite, then the groups \({\tilde{\varGamma }}\) from Theorem 1 would provide a counterexample to this.

In fact one can show as a consequence of the results proved here the following slightly more precise result.

### Corollary 2

Assume that every Gromov–hyperbolic group is residually finite. If \(G/\mathbb {Q}\) is a simple, simply connected group of real rank 1 then for every positive integer *n*, the congruence kernel of *G* has a subquotient isomorphic to \(\mathbb {Z}/n\).

## 2 Statement of results

- 1.
*G*is (algebraically) simply connected; - 2.
*G*has positive real rank (i.e. \(G(\mathbb {R})\) is not compact, and arithmetic subgroups of*G*are infinite); - 3.
The congruence kernel of \(G/\mathbb {Q}\) is finite (and hence conjecturally the real rank of

*G*is at least 2).

*G*is absolutely simple.

We’ll show that Theorem 1 is a consequence of the following result.

### Theorem 2

*n*there is a subgroup \(\varDelta \) of finite index in \(\varGamma \) and a central extension

### Proof of Theorem 1

*A*is the induced representation \(A=\mathrm {ind}_\varDelta ^\varGamma (\mathbb {Z}/n)\). We’ll write \(\varSigma \) for the image of \(\sigma \) in \(H^2(\varGamma ,A)\). Corresponding to the cohomology class \(\varSigma \), there is a (non-central) extension \({\tilde{\varGamma }}\) of \(\varGamma \) by

*A*. These two group extensions are related by the following commutative diagram: Suppose for the sake of argument that \({\tilde{\varGamma }}\) is residually finite. Hence the subgroup \(\mathrm {pr}^{-1}\varDelta \) is residually finite. There is therefore a subgroup \(\varPhi \subset \mathrm {pr}^{-1}(\varDelta )\) of finite index, such that \(\varPhi \cap A\) is trivial. The image of \(\varPhi \) in \({\tilde{\varDelta }}\) is then a subgroup of finite index in \({\tilde{\varDelta }}\), whose intersection with \(\mathbb {Z}/n\) is trivial. This is a contradiction. \(\square \)

### 2.1 Some refinements of Theorem 2

Let \(G/\mathbb {Q}\) be simple, simply connected, and have real rank at least 1. Furthermore assume that the congruence kernel of *G* is finite (and hence, conjecturally at least, the real rank of *G* is at least 2). Fix an arithmetic subgroup \(\varGamma \) of *G*.

### Theorem 3

For every positive integer *n*, there are infinitely many elements of order *n* in \({\bar{H}}^{2}(\mathbb {Z}/n)\).

*arithmetic completion*of the group \(G(\mathbb {Q})\), i.e.

*S*be a finite set of prime numbers. By an

*S-arithmetic level*, we shall mean an open subgroup

*L*of \({\widehat{G(\mathbb {Q})}}\) of the form

*L*is open in \({\widehat{G(\mathbb {Q})}}\).

### Theorem 4

Let *L* be an *S*-arithmetic level in \({\widehat{G(\mathbb {Q})}}\) for some finite set of primes *S*. For every positive integer *n*, there are infinitely many elements of order *n* in \({\bar{H}}^{2}(\mathbb {Z}/n)^L\).

Theorem 4 will be proved in Sect. 4. The proof requires a technical result on the cohomology of finite groups of Lie type, which is proved in Sect. 5. By modifying the argument slightly, one can also prove the following result.

### Theorem 5

Let *n* be a positive integer. Then there are infinitely many elements \(\sigma \) of order *n* in \({\bar{H}}^2(\mathbb {Z}/n)\) with the following property. There is a prime number *p* depending on \(\sigma \), such that for all primes \(q\ne p\) the element \(\sigma \) is fixed by \(\mathrm {pr}^{-1}\left( G(\mathbb {Q}_q) \right) \).

### 2.2 Virtual lifting to characteristic zero

*l*be a prime number. Any central extension of \(\varGamma \) by \(\mathbb {Z}/l^{t+1}\) gives rise to a central extension by \(\mathbb {Z}/l^t\). We’ll say that the extension of \(\varGamma \) by \(\mathbb {Z}/l^r\)

*virtually lifts to characteristic zero*if for every \(t>r\) there is a arithmetic subgroup \(\varDelta _t\) of \(\varGamma \) and a central extension of \(\varDelta _t\) by \(\mathbb {Z}/l^t\), such that the extensions fit into a commutative diagram of the following form. Here the map \(\mathbb {Z}/l^{t} \rightarrow \mathbb {Z}/l^r\) is the usual reduction map, and the map \(\varDelta _t \rightarrow \varGamma \) is the inclusion.

*complex type*if \(G_i\) is the restriction of scalars of a group defined over \(\mathbb {C}\), or equivalently if \(G_i(\mathbb {C})\) is a product of two simple groups; otherwise we say that \(G_i\) is of real type. We’ll write \(b_\mathbb {R}\) for the number of simple factors of \(G(\mathbb {R})\) of real type and \(b_\mathbb {C}\) for the number of simple factors of \(G(\mathbb {R})\) of complex type. We’ll also write

*m*for the dimension of the centre of a maximal compact subgroup \(K_\infty \subset G(\mathbb {R})\).

### Theorem 6

*c*. More precisely,

*c*is in the range

*m*are the integers defined above. In particular \({\bar{H}}^{2}({\mathbb {Z}_l})\) is non-zero.

For comparison, we note that the construction of Deligne implies the bound \(c \ge m\); this is because \(\pi _1(G(\mathbb {R}))\) has a finite index subgroup isomorphic to \(\mathbb {Z}^m\).

As an easy consequence of the theorem, we obtain the following:

### Corollary 3

Let \(G/\mathbb {Q}\) be simple and simply connected with finite congruence kernel. There is a subgroup of \({\bar{H}}^2(\mathbb {Z}/l^t)^{\widehat{G(\mathbb {Q})}}\) isomorphic to \((\mathbb {Z}/l^t)^{c}\), all of whose elements virtually lift to characteristic zero, where *c* is the positive integer in Theorem 6.

Theorem 6 and its corollary will be proved in section 6. The proof requires a result on the cohomology of compact symmetric spaces, which is proved in the appendix.

### Remark 1

We stress that Theorem 6 implies \({\bar{H}}^2({\mathbb {Z}_l})^{{\widehat{G(\mathbb {Q})}}}\) is non-zero even in cases where \(H^2(\varGamma ,\mathbb {C})=0\) for all arithmetic subgroups \(\varGamma \) of \(G(\mathbb {Q})\). This happens when *G* has large real rank and the symmetric space associated to *G* has no complex structure, for example when \(G=\mathrm {SL}_5/\mathbb {Q}\). The extensions constructed by the method of Deligne exist only in the case \(m>0\); our result shows that \({\bar{H}}^2({\mathbb {Z}_l})^{\widehat{G(\mathbb {Q})}}\) is non-zero even in cases where \(m=0\).

### Remark 2

The author suspects that \(\mathrm {rank}_{\mathbb {Z}_l}\left( \bar{H}^{2}({\mathbb {Z}_l})^{{\widehat{G(\mathbb {Q})}}}\right) =b_\mathbb {R}+b_\mathbb {C}+m\). Proving this would amount to showing that the restriction map \(H^3_\mathrm {cts}(G(\mathbb {Q}_l),\mathbb {Q}_l) \rightarrow H^3(G(\mathbb {Q}),\mathbb {Q}_l)\) is surjective. The evidence for this is very slight, but we note that \(\dim H^3_\mathrm {cts}(G(\mathbb {Q}_l),\mathbb {Q}_l)\) is at least twice as big as \(\dim H^3(G(\mathbb {Q}),\mathbb {Q}_l)\).

As an example, consider the case \(G=\mathrm {Res}^k_\mathbb {Q}(SL_{\ge 3}/k)\), where *k* is an imaginary quadratic field. In this case \(m=0\), \(b_\mathbb {R}=0\) and \(b_\mathbb {C}=1\), so our result implies that the rank *c* is either 1 or 2. In this case \(H^3_\mathrm {cts}(G(\mathbb {Q}_l),\mathbb {Q}_l)\) is 2-dimensional and \(H^3(G(\mathbb {Q}),\mathbb {Q}_l)\) is 1-dimensional (see Sect. 6.3), so the restriction map is either surjective or zero. If the restriction map is non-zero, then the rank is 1. One might expect to prove that the rank is 1 by evaluating an appropriate *l*-adic Borel regulator; however the author has not done this in any case.

*r*,

*s*). The congruence subgroup property for such groups was established by Kneser [12].

Values of the rank of \(\bar{H}^2(\mathbb {Z}_l)^{\widehat{G(\mathbb {Q})}}\)

| | \(b_\mathbb {R}\) | \(c = \mathrm {rank}_{{\mathbb {Z}_l}}\left( {\bar{H}}^{2}({\mathbb {Z}_l})^{{\widehat{G(\mathbb {Q})}}}\right) \) | |
---|---|---|---|---|

\(\mathrm {SL}_{n}/\mathbb {Q}\) | (\(n \ge 3\)) | 0 | 1 | 1 |

\(\mathrm {Sp}_{2n}/\mathbb {Q}\) | (\(n \ge 2\)) | 1 | 1 | 2 |

\(\mathrm {Spin}(r,s)\) | (\(r \ge s \ge 3\)) | 0 | 1 | 1 |

\(\mathrm {Spin}(r,2)\) | (\(r \ge 3\)) | 1 | 1 | 2 |

\(\mathrm {Spin}(2,2)\) | 2 | 2 | 4 | |

\(\mathrm {Res}^{k}_{\mathbb {Q}}(\mathrm {SL}_{n}/k)\) | (\(n \ge 3\), | 0 | \([k:\mathbb {Q}]\) | \([k:\mathbb {Q}]\) |

\(\mathrm {Res}^{k}_{\mathbb {Q}}(\mathrm {SL}_{2}/k)\) | ( | \([k:\mathbb {Q}]\) | \([k:\mathbb {Q}]\) | \(2[k:\mathbb {Q}]\) |

\(\mathrm {Res}^{k}_{\mathbb {Q}}(\mathrm {Sp}_{2n}/k)\) | ( | \([k:\mathbb {Q}]\) | \([k:\mathbb {Q}]\) | \(2[k:\mathbb {Q}]\) |

The case \(\mathrm {SL}_{2}/\mathbb {Q}\) and its forms of rank 0 are not included in the table. This is because these groups have infinite congruence kernel, and indeed for these groups we have \({\bar{H}}^{2}(\mathbb {Z}/n)=0\) and \({\bar{H}}^2({\mathbb {Z}_l})=0\).

## 3 Background material

### 3.1 Continuous cohomology

We shall make use of the continuous cohomology groups \(H^\bullet _\mathrm {cts}(G,A)\), where *G* is a topological group and *A* is an abelian topological group, which is a *G*-module via a continuous action \(G \times A \rightarrow A\).

In all cases under consideration here, the group *G* will be metrizable, locally compact, totally disconnected, separable and \(\sigma \)-compact. The coefficient group *A* will always be Polonais (a topological group is Polonais if its topology admits a separable complete metric; see page 3 of [18]). Under these restriction, the continuous cohomology groups defined in [7] (based on continuous cocycles) are the same as those defined in [16, 17, 18] based on Borel measurable cocycles. This is proved in Theorem 1 of [25].

If *A* is a continuous *H*-module for some closed subgroup *H* of *G*, then we shall write \(\mathrm {ind}_H^G(A)\) for the induced module, consisting of all continuous functions \(f:G \rightarrow A\) satisfying \(f(hg)=h\cdot f(g)\) for all \(g\in G\) and \(h\in H\). This agrees with the notation of [7] but not [16, 17, 18]. The following version of Shapiro’s lemma holds for these induced representations.

### Theorem 7

*H*be a closed subgroup of

*G*, where

*G*satisfies the conditions above. For any continuous

*H*-module

*A*, there is a canonical isomorphism of topological groups:

### Proof

This follows Propositions 3 and 4 of [7] in view of the remark following Proposition 4. \(\square \)

We shall also make frequent use of the following.

### Theorem 8

*H*be a closed normal subgroup of a group

*G*, where

*G*satisfies the conditions above, and let

*A*be a continuous Polonais representation of

*G*. If the groups \(H^\bullet (H,A)\) are all Hausdorff, then there is a first quadrant spectral sequence converging to \(H^\bullet (G,A)\), with \(E_2\) sheet given by

### Proof

This follows from Theorem 9 of [18] in all cases under consideration. \(\square \)

### 3.2 The derived functor of projective limit

*projective system*, we shall mean a sequence of abelian groups \(A_{t}\), indexed by \(t\in \mathbb {N}\), and connected by group homomorphisms as follows:

The projective system \((A_{t})\) is said to satisfy the *Mittag–Leffler property* if for every \(t\in \mathbb {N}\), there is a \(j\in \mathbb {N}\) with the property that for all \(k>j\) the image of \(A_{k}\) in \(A_{t}\) is equal to the image of \(A_{j}\) in \(A_{t}\). For example, if the Abelian groups \(A_{t}\) are all finite then the projective system has the Mittag–Leffler property. Similarly, if the groups \(A_t\) are all finite dimensional vector spaces connected by linear maps, then the projective system satisfies the Mittag–Leffler condition.

### Proposition 1

(Proposition 3.5.7 of [24]) If the projective system \((A_{t})\) satisfies the Mittag–Leffler condition then \( \left( \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array}\right) ^{1} A_{t} = 0 \).

### Theorem 9

*S*-arithmetic subgroups. As before, we let \(G/\mathbb {Q}\) be a simple, simply connected algebraic group, and \(K_f=\prod _p K_p\) a compact open subgroup of \(G(\mathbb {A}_{f})\). We shall write \(\varGamma \) for the arithmetic group \(G(\mathbb {Q}) \cap K_f\). More generally, if

*S*is a finite set of prime numbers, then we use the notation \(\varGamma ^S\) for the corresponding

*S*-arithmetic group, i.e.

### Proposition 2

For any field \(\mathbb {F}\), we have \( H^\bullet (G(\mathbb {Q}), \mathbb {F}) = \begin{array}{c} \lim \\ {\mathop {\scriptstyle S}\limits ^{\textstyle \leftarrow }} \end{array} H^\bullet (\varGamma ^S,\mathbb {F}). \) In the case \(\mathbb {F}=\mathbb {C}\) we have \( H^\bullet (G(\mathbb {Q}),\mathbb {C}) = H^\bullet (\mathfrak {g},\mathfrak {k},\mathbb {C}), \) where \(H^\bullet (\mathfrak {g},\mathfrak {k},\mathbb {C})\) are the relative Lie algebra cohomology groups studied in [5].

### Proof

In the case \(\mathbb {F}=\mathbb {C}\), the theorem of [2] implies that \(H^r(\varGamma ^S,\mathbb {C})=H^r(\mathfrak {g},\mathfrak {k},\mathbb {C})\) whenever *S* contains more than *r* primes. Hence the projective limit (over *S*) is in this case \(H^r(\mathfrak {g},\mathfrak {k},\mathbb {C})\). \(\square \)

### 3.3 The congruence kernel

*arithmetic subgroup*of

*G*is any subgroup of \(G(\mathbb {Q})\), which is commensurable with a congruence subgroup. The

*arithmetic completion*\({\widehat{G(\mathbb {Q})}}\) is defined to be the completion of \(G(\mathbb {Q})\) with respect to the arithmetic subgroups of

*G*, i.e.

*congruence kernel*\(\mathrm {Cong}(G)\) is defined to be the kernel of this map, so we have a short exact sequence:

*G*is a congruence subgroup. If \(G(\mathbb {R})\) is simply connected as an analytic group, then the congruence kernel is never trivial, but may still be finite. It has been conjectured by Serre [20], that the congruence kernel is finite if and only if each simple factor of

*G*over \(\mathbb {Q}\) has real rank at least 2. In the case that \(\mathrm {Cong}(G)\) is finite, it is known that \(\mathrm {Cong}(G)\) is contained in the centre of \({\widehat{G(\mathbb {Q})}}\), and is a cyclic group.

## 4 Proof of Theorem 4

In this section, we assume that the group \(G/\mathbb {Q}\) is a simple, simply connected algebraic group with positive real rank. We shall also assume that the congruence kernel \(\mathrm {Cong}(G)\) is finite. Hence, conjecturally that the real rank of *G* is at least 2.

### 4.1 The groups \(\mathcal {C}(L,\mathbb {Z}/n)\)

Let *L* be an open subgroup of the arithmetic completion \({\widehat{G(\mathbb {Q})}}\). We shall write \({\varGamma (L)}\) for the group \(G(\mathbb {Q})\cap L\). Since \(G(\mathbb {Q})\) is dense in \({\widehat{G(\mathbb {Q})}}\), it follows that \({\varGamma (L)}\) is dense in *L*. If *L* is compact and open then \({\varGamma (L)}\) is an arithmetic group and *L* is its profinite completion. If *L* is an *S*-arithmetic level, then \({\varGamma (L)}\) is an *S*-arithmetic group.

We shall write \(\mathcal {C}(L,\mathbb {Z}/n)\) for the group of continuous functions \(f:L \rightarrow \mathbb {Z}/n\). We regard \(\mathcal {C}(L,\mathbb {Z}/n)\) as a \({\varGamma (L)}\times L\)-module, in which (for the sake of argument) \({\varGamma (L)}\) acts by left-translation and *L* acts by right-translation. We regard \({\varGamma (L)}\) as a discrete topological group, and *L* as a topological group with the subspace topology from \({\widehat{G(\mathbb {Q})}}\). We do not assume that elements of \(\mathcal {C}(L,\mathbb {Z}/n)\) are uniformly continuous, and so the action of *L* is not smooth unless *L* is compact. The action is continuous, where \(\mathcal {C}(L,\mathbb {Z}/n)\) is equipped with the compact–open topology.

### Proposition 3

*L*of \({\widehat{G(\mathbb {Q})}}\), there is a canonical isomorphism of

*L*-modules:

### Proof

*L*. Let

*K*be an open subgroup of

*L*. As a \({\varGamma (L)}\)-module, we have

*L*is compact and open. Under this assumption, we have (as \({\varGamma (L)}\)-modules):

*L*to be compact, then \(\mathcal {C}(L,\mathbb {Z}/n)\) is discrete, and therefore the group \(H^\bullet ({\varGamma (L)},\mathcal {C}(L,\mathbb {Z}/n))\) is discrete. \(\square \)

### Lemma 1

We have \( H^{0}_{\mathrm {cts}}(L, \mathcal {C}(L,\mathbb {Z}/n)) = \mathbb {Z}/n\) and \(H^s_\mathrm {cts}(L,\mathcal {C}(L,\mathbb {Z}/n))=0\) for \(s>0\). In particular the groups \(H^{s}_{\mathrm {cts}}(L, \mathcal {C}(L,\mathbb {Z}/n))\) are Hausdorff.

### Proof

As a continuous *L*-module, we have \( \mathcal {C}(L,\mathbb {Z}/n)= \mathrm {ind}_{1}^{L} (\mathbb {Z}/n). \) The result follows from this using Shapiro’s Lemma. \(\square \)

### Proposition 4

Let *L* be any open subgroup of \({\widehat{G(\mathbb {Q})}}\). Then there is a first quadrant spectral sequence with \(E_2^{r,s} = H^{r}_{\mathrm {cts}}(L, \bar{H}^{s}(\mathbb {Z}/n))\) which converges to \(H^{r+s}({\varGamma (L)}, \mathbb {Z}/n)\).

### Proof

### 4.2 Low degree terms

We shall now describe some of the low degree terms of the spectral sequence of Proposition 4.

### Lemma 2

### Proof

### Theorem

Let *S* be a finite set of prime numbers and let *L* be an *S*-arithmetic level. Then the group \({\bar{H}}^2(\mathbb {Z}/n)^L\) contains infinitely many elements of order *n*.

### Proof

*L*be an

*S*-arithmetic level. In this case the group \({\varGamma (L)}\) is an

*S*-arithmetic group. By the theory of the Borel–Serre compactification, there is a resolution of \(\mathbb {Z}\) as a \({\varGamma (L)}\)-module consisting of finitely generated \(\mathbb {Z}[{\varGamma (L)}]\)-modules. This implies that the cohomology groups \(H^r({\varGamma (L)},\mathbb {Z}/n)\) are all finite. In view of this, the sequence in Equation 2 has the form

*n*.

*p*will be called a

*tame*prime if it satisfies all of the following conditions:

- 1.
*p*is not in the finite set*S*; - 2.
*p*is not a factor of \(|\mathrm {Cong}(G)|\); - 3.
*p*is not a factor of*n*; - 4.
*G*is unramified over \(\mathbb {Q}_p\). - 5.
The group \(K_p\) is a maximal hyperspecial compact open subgroup of \(G(\mathbb {Q}_p)\) (see [23]). This implies that if we let \(K_p^0\) be the maximal pro-

*p*normal subgroup of \(K_p\), then the quotient \(G(\mathbb {F}_p)=K_p/K_p^0\) is a product of some of the simply connected finite Lie groups described in [22]. - 6.
\(H^r(G(\mathbb {F}_p),\mathbb {Q}/\mathbb {Z})=0\) for \(r=1,2\). We recall from [22] that this condition is satisfied for all but finitely many of the groups \(G(\mathbb {F}_p)\).

*p*, we shall write \(K_p^*\) for a lift of \(K_p\) to \({\widehat{G(\mathbb {Q})}}\); note that such a lift exists and is unique by conditions (2) and (6). The group

*L*contains the following subgroup

*L*. Hence by the Künneth formula, \(H^3_\mathrm {cts}(K_\mathrm {tame}, \mathbb {Z}/n)\) is a direct summand of \(H^3_\mathrm {cts}(L, \mathbb {Z}/n)\). It is therefore sufficient to prove that \(H^3_\mathrm {cts}(K_\mathrm {tame},\mathbb {Z}/n)\) contains infinitely many elements of order

*n*.

*p*, we may identify \(H^\bullet _\mathrm {cts}(K_p,\mathbb {Z}/n)\) with \(H^\bullet (G(\mathbb {F}_p),\mathbb {Z}/n)\). To prove the theorem, it is therefore sufficient so show that there are infinitely many tame primes

*p*, such that \(H^3(G(\mathbb {F}_p),\mathbb {Z}/n)\) contains an element of order

*n*. This follows from Theorem 10, which will be proved in the next section. \(\square \)

## 5 A lemma on the cohomology of finite Lie groups

In this section we shall prove Theorem 10, which completes the proof of Theorem 4.

Before stating the theorem, we note that if *G* is an algebraic group over \(\mathbb {Q}\), then we may write *G* in the form \(\mathscr {G}\times _\mathbb {Z}\mathbb {Q}\), for some group scheme \(\mathscr {G}\) over \(\mathbb {Z}\). The group \(\mathscr {G}(\mathbb {F}_p)\) depends on the choice of \(\mathscr {G}\), not just on *G*. Nevertheless if we alter the group scheme \(\mathscr {G}\) then only finitely many of the groups \(\mathscr {G}(\mathbb {F}_p)\) will change. Because of this, the following statement makes sense, where we are writing \(G(\mathbb {F}_p)\) in place of \(\mathscr {G}(\mathbb {F}_p)\) for some fixed choice of \(\mathscr {G}\).

### Theorem 10

Let \(G/\mathbb {Q}\) be a simple, simply connected algebraic group. For every positive integer *n* there are infinitely many prime numbers *p*, such that \(H^3(G(\mathbb {F}_p),\mathbb {Z}/n)\) contains an element of order *n*.

I assume this sort of result is known to experts, and many special cases are consequences of results in algebraic K-theory (for example the results of [19] imply the case \(\mathrm {SL}_r\)).

*T*be a subgroup of a finite group

*G*, and let

*A*be a

*G*-module. We shall write \(\mathrm {Rest}^G_T\) and \(\mathrm {CoRest}^T_G\) for the restriction and corestriction maps between \(H^\bullet (G,A)\) and \(H^\bullet (T,A)\). A cohomology class \(\sigma \in H^r(T,A)\) is called

*invariant*if for every \(g\in G\),

### Proposition 5

*T*be a subgroup of a finite group

*G*. If \(\sigma \in H^\bullet (T,A)\) is an invariant cohomology class. Then

As a corollary to this, we note the following.

### Corollary 4

Let *T* be a subgroup of a finite group *G*. Let *d* be a positive integer and *l* a prime number, such that \(\big |[G:T]\big |_l=\big |d\big |_l\). If \(H^r(T,\mathbb {Z})\) contains an invariant class of order \(dl^t\) then \(H^r(G,\mathbb {Z})\) contains an element of order \(l^t\).

### Proof

Let \(\tau = \mathrm {CoRest}^T_G(\sigma )\), where \(\sigma \) is the invariant class on *T* of order \(dl^t\). By Proposition 5, the restriction of \(\tau \) to *T* has order \(\frac{dl^t}{\gcd ( dl^t, [G:T])}\). The condition on *d* implies that the order of \(\mathrm {Rest}^G_T(\tau )\) is a multiple of \(l^t\). Hence the order of \(\tau \) is a multiple of \(l^t\), so some multiple of \(\tau \) has order \(l^t\). \(\square \)

In order to apply the corollary, it will be useful to note the following.

### Lemma 3

*l*be a prime number, and let

*x*be an integer such that \(x \equiv 1 \bmod 2l\). Then for every integer

*d*we have

### Proof

*l*-adic logarithm function \(\log _l\) converges on the multiplicative group \(1+2l\mathbb {Z}_l\). If \(\log _l\) converges at an element

*x*, then we have \(|\log (x)|_l = | x-1 |_l\). Our congruence condition implies that \(\log _l(x)\) and \(\log _l(x^d)\) both converge, so we have

### Proof of Theorem 10

By the Chinese remainder theorem, it is sufficient to prove the theorem in the case \(n=l^t\), where *l* is a prime number.

*G*over \(\mathbb {Z}\), and let

*k*be a number field such that \(\mathscr {G}\) splits over \(\mathscr {O}_k\). Let \(\mathscr {T}\) be a maximal torus in \(\mathscr {G}\), defined and split over \(\mathscr {O}_k\). Let

*P*be the lattice of algebraic characters \(\mathscr {T}\rightarrow \mathrm {GL}_1/\mathscr {O}_k\). The roots of \(\mathscr {G}\) with respect to \(\mathscr {T}\) are elements of the lattice

*P*. Consider the element

*P*with a group of characters of the Lie algebra \(\mathfrak {t}\) of \(\mathscr {T}\), then we may similarly identify elements of \(\mathrm {Sym}^2(P)\) with quadratic forms on \(\mathfrak {t}\). The element

*Q*corresponds to the restriction of the Killing form to \(\mathfrak {t}\). Therefore

*Q*is non-zero.

Let *e* be the largest positive integer, such that *Q* is a multiple of *e* in the lattice \(\mathrm {Sym}^2 (P)\). Also let \(d_1,\ldots , d_r\) be the degrees of the basic polynomial invariants of the Weyl group of *G* / *k* (where *r* is the rank of *G* / *k*). The smallest of these degrees is \(d_1=2\), and the others depend on the root system (see [22]). By extending the number field *k* if necessary, we may assume that *k* contains a primitive root of unity of order \(d_1\cdots d_r\cdot e\cdot n\). By the Chebotarev density theorem, there are infinitely many prime numbers which split in *k*; we’ll show that each of these prime numbers has the desired property.

From now on we fix a prime number *p* which splits in *k*, and we are attempting to show that \(H^3(\mathscr {G}~(\mathbb {F}_p),\mathbb {Z}/n)\) contains an element of order *n*. By abusing notation slightly we shall write \(G(\mathbb {F}_p)\) for the group \(\mathscr {G}~(\mathbb {F}_p)\). We may identify \(G(\mathbb {F}_p)\) with \(\mathscr {G}~(\mathscr {O}_k/\mathfrak {p})\) for some prime ideal \(\mathfrak {p}\) above *p*). We shall also write \(T(\mathbb {F}_p)\) for the subgroup \(\mathscr {T}~(\mathscr {O}_k/\mathfrak {p})\).

Identifying \(H^3(G(\mathbb {F}_p),\mathbb {Z}/n)\) with the *n*-torsion in \(H^4(G(\mathbb {F}_p),\mathbb {Z})\), we see that it’s sufficient to prove there is an element of order *n* in \(H^4(G(\mathbb {F}_p),\mathbb {Z})\).

*N*is the number of positive roots;

*r*is the rank and \(d_1,\ldots ,d_r\) are the degrees of the fundamental invariants of the Weyl group. Note also that since

*T*is a split torus of rank

*r*, we have

*p*splits in

*k*and

*k*contains an primitive 2

*l*-th root of unity (because \(d_1=2\)), we have \(p \equiv 1 \bmod 2l\). Hence by Lemma 3,

*p*.

*P*is the lattice of algebraic characters of

*T*.

^{1}. From this, we see that \(H^4(T(\mathbb {F}_p),\mathbb {Z})\otimes (\mathbb {F}_p^\times )^{\otimes 2}\) contains as a subgroup the group

*G*with respect to

*T*. The element

*q*is evidently in the subgroup \(\mathrm {Sym}^2(P)/(p-1)\). Equivalently, we can regard

*q*as the quadratic function \(q:T(\mathbb {F}_p) \rightarrow (\mathbb {F}_p^\times )^{\otimes 2}\) defined by

*g*is an element of \(G(\mathbb {F}_p)\) and suppose that both

*t*and \(g^{-1}tg\) are in \(T(\mathbb {F}_p)\). To show that

*q*is an invariant class, we must show that \(q(t)=q(g^{-1}tg)\). Evidently we have

*t*on the Lie algebra \(\mathfrak {g}\otimes \mathbb {F}_p\). These eigenvalues are the same as those of \(g^{-1} t g\), and so the numbers \(\alpha (t)\) are the same (possibly in a different order) as the numbers \(\alpha (g^{-1} t g)\). From this it follows that \(q(t^g)=q(t)\), so

*q*is an invariant class in \(H^4(T(\mathbb {F}_p),\mathbb {Z})\otimes (\mathbb {F}_p^\times )^{\otimes 2}\).

It remains to determine the order of *q* in \(H^4(T(\mathbb {F}_p),\mathbb {Z})\otimes (\mathbb {F}_p^\times )^{\otimes 2}\), or equivalently the order of *q* in the subgroup \(\mathrm {Sym}^2(P)/(p-1)\). By definition, *q* is the the reduction modulo \(p-1\) of the element \(Q\in \mathrm {Sym}^2(P)\). We defined *e* to be the largest integer such that *Q* is a multiple of *e*. Since we are assuming that \(p\equiv 1 \bmod e\), the order of *q* in \(\mathrm {Sym}^2(P)/(p-1)\) is precisely \(\frac{p-1}{e}\).

To summarize, we have shown that \(H^4(T(\mathbb {F}_p),\mathbb {Z})\) has an invariant element of order \(\frac{p-1}{e}\). Therefore \(H^4(G(\mathbb {F}_p),\mathbb {Z})\) has an element of order \(\frac{p-1}{d_1 \cdots d_r \cdot e}\). Since \(p \equiv 1 \bmod (d_1 \cdots d_r \cdot e \cdot n)\) it follows that \(H^4(G(\mathbb {F}_p),\mathbb {Z})\) has an element of order *n*. \(\square \)

## 6 Proof of Theorem 6

### 6.1 The groups \({\bar{H}}^2({\mathbb {Z}_l})\)

*L*be an open subgroup of the arithmetic completion \({\widehat{G(\mathbb {Q})}}\), and we shall now fix a prime number

*l*. We introduce a new module

### Proposition 6

- 1.
The groups \({\bar{H}}^\bullet ({\mathbb {Z}_l})\) do not depend on the open subgroup

*L*in their definition (Equation 3). - 2.
\({\bar{H}}^0({\mathbb {Z}_l})={\mathbb {Z}_l}\),

- 3.
\({\bar{H}}^1({\mathbb {Z}_l})=0\),

- 4.
\( {\bar{H}}^2({\mathbb {Z}_l}) = \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} {\bar{H}}^2(\mathbb {Z}/l^t). \)

- 5.
The group \({\bar{H}}^2({\mathbb {Z}_l})\) is torsion-free and contains no non-zero divisible elements.

- 6.
For any open subgroup

*L*of \({\widehat{G(\mathbb {Q})}}\) we have \( \bar{H}^2({\mathbb {Z}_l})^L = \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} \left( {\bar{H}}^2(\mathbb {Z}/l^t)^L \right) \).

### Proof

- 1.
Suppose

*M*is an open subgroup of*L*. Then we have an isomorphism of \({\varGamma (L)}\)-modules \(\mathcal {C}(L,{\mathbb {Z}_l})= \mathrm {ind}_M^L \mathcal {C}(M,{\mathbb {Z}_l})\). The result follows from this by Shapiro’s Lemma (Theorem 7). - 2.
Since \({\varGamma (L)}\) is dense in

*L*, it follows that the \({\varGamma (L)}\)-invariant continuous functions on*L*are constant. This shows that \({\bar{H}}^0({\mathbb {Z}_l})={\mathbb {Z}_l}\). - (3,4)For each \(r>0\) we have by Theorem 9 a short exact sequenceIn the notation of the previous section, we have$$\begin{aligned}&0 \rightarrow \left( \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} \right) ^1 H^{r-1}(\varGamma (L),\mathcal {C}(L,\mathbb {Z}/l^t)) \rightarrow {\bar{H}}^{r}({\mathbb {Z}_l})\\&\qquad \quad \quad \rightarrow \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} H^{r}(\varGamma (L),\mathcal {C}(L,\mathbb {Z}/l^t)) \rightarrow 0. \end{aligned}$$By Lemma 2, \({\bar{H}}^0(\mathbb {Z}/l^t)=\mathbb {Z}/l^t\) and \(\bar{H}^1(\mathbb {Z}/l^t)=0\). Both of these projective systems consist of finite groups, so satisfy the Mittag–Leffler condition. Therefore \(\left( \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} \right) ^1\) vanishes on both of them. As a result of this we have for \(r=1,2\):$$\begin{aligned} 0 \rightarrow \left( \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} \right) ^1 {\bar{H}}^{r-1}(\mathbb {Z}/l^t) \rightarrow {\bar{H}}^{r}({\mathbb {Z}_l}) \rightarrow \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} {\bar{H}}^{r}(\mathbb {Z}/l^t) \rightarrow 0. \end{aligned}$$In particular \({\bar{H}}^1({\mathbb {Z}_l})=0\).$$\begin{aligned} {\bar{H}}^{r}({\mathbb {Z}_l}) = \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} {\bar{H}}^{r}(\mathbb {Z}/l^t). \end{aligned}$$
- (5)Consider the short exact sequence of modules:This gives the exact sequence in cohomology$$\begin{aligned} 0 \rightarrow \mathcal {C}(L,{\mathbb {Z}_l}){\mathop {\rightarrow }\limits ^{\times l^t}} \mathcal {C}(L,{\mathbb {Z}_l})\rightarrow \mathcal {C}(L,\mathbb {Z}/l^t)\rightarrow 0. \end{aligned}$$We already saw in Lemma 2 that \({\bar{H}}^1(\mathbb {Z}/l^t)=0\). This shows that \({\bar{H}}^2({\mathbb {Z}_l})\) is torsion-free. Suppose \(\sigma \) is a divisible element in \({\bar{H}}^2({\mathbb {Z}_l})\). Then the image of \(\sigma \) in \({\bar{H}}^2(\mathbb {Z}/l^t)\) is a divisible element for each$$\begin{aligned} \cdots \rightarrow {\bar{H}}^1(\mathbb {Z}/l^t) \rightarrow {\bar{H}}^2({\mathbb {Z}_l}) {\mathop {\rightarrow }\limits ^{\times l^t}} {\bar{H}}^2({\mathbb {Z}_l}) \rightarrow \cdots . \end{aligned}$$
*t*. Since \(\bar{H}^2(\mathbb {Z}/l^t)\) is a \(\mathbb {Z}/l^t\)-module, the image of \(\sigma \) in \(\bar{H}^2(\mathbb {Z}/l^t)\) must be zero. By (4) it follows that \(\sigma =0\). - (6)
This follows because the functor \( \begin{array}{c} \lim \\ {\mathop {\scriptstyle t}\limits ^{\textstyle \leftarrow }} \end{array} \) commutes with the functor \(-^L\) of

*L*-invariant elements. \(\square \)

### Proposition 7

*S*-arithmetic level \(L \subset {\widehat{G(\mathbb {Q})}}\), there is an exact sequence as follows:

### Remark 3

It is tempting to suggest that the exact sequence of the proposition follows from a spectral sequence of the form \(H^r_\mathrm {cts}(L,\bar{H}^s({\mathbb {Z}_l})) \implies H^{r+s}({\varGamma (L)},{\mathbb {Z}_l})\), which would be proved in the same way as in the finite coefficient case (Proposition 4). Unfortunately this is not quite so simple. The problem is that the groups \({\bar{H}}^r({\mathbb {Z}_l})\) will probably not be Hausdorff for \(r \ge 3\), and so there is no off-the-shelf spectral sequence for us to use. Admittedly we could truncate at \({\bar{H}}^2({\mathbb {Z}_l})\) to obtain a spectral sequence with three rows, or we could try to work with the more general spectral sequence constructed in [8]. Instead we’ve gone for a more elementary approach, and we prove the exact sequence of the proposition by taking the projective limit of such exact sequences in the finite coefficient cases.

### Proof

*S*-arithmetic group, the cohomology groups \(H^\bullet ({\varGamma (L)},\mathbb {Z}/l^t)\) are all finite. From the Equation 4 it follows that \(H^2_\mathrm {cts}(L,\mathbb {Z}/l^t)\) and \(A_t\) are both finite, and hence

*S*-arithmetic group, the groups \(H^\bullet ({\varGamma (L)},\mathbb {Z}/l^t)\) must be finite. Furthermore, the spectral sequence in Eq. 1 shows that the groups \(H^1_\mathrm {cts}(L,\mathbb {Z}/l^t)\) and \(H^2_\mathrm {cts}(L,\mathbb {Z}/l^t)\) are also finite. As a result, all of these projective systems satisfy the Mittag–Leffler condition, so we have:

### 6.2 The groups \(H^\bullet _\mathrm {cts}(L,{\mathbb {Z}_l})\)

We shall next concentrate on the continuous cohomology groups in the exact sequence of Proposition 7.

*L*is an

*S*-arithmetic level for some finite set of primes

*S*. Recall that this means

*L*is the pre-image in \({\widehat{G(\mathbb {Q})}}\) of an open subgroup of \(G(\mathbb {A}_{f})\) of the form

*p*a

*tame prime*if it satisfies all of the following conditions:

- 1.
\(p \not \in S\).

- 2.
\(p \ne l\).

- 3.
*G*is unramified over \(\mathbb {Q}_p\). - 4.
\(K_p\) is a maximal hyperspecial compact open subgroup of \(G(\mathbb {Q}_p)\). This implies that if we let \(K_p^0\) be the maximal pro-

*p*normal subgroup of \(K_p\), then the group \(G(\mathbb {F}_p)=K_p/K_p^0\) is a product of some of the simply connected finite groups of Lie type described in detail in [22]. - 5.
\(H^r(G(\mathbb {F}_p),\mathbb {Q}/\mathbb {Z})=0\) for \(r=1,2\). We recall from [22] that this condition is satisfied for all but finitely many of the groups \(G(\mathbb {F}_p)\).

*p*we have \(H^\bullet _\mathrm {cts}(K_p,\mathbb {Z}/l^t)= H^\bullet (G(\mathbb {F}_p),\mathbb {Z}/l^t)\) by condition (2).

*W*for the set of primes not in

*S*which are not tame. The group \(L/\mathrm {Cong}(G)\) decomposes in the form

### Lemma 4

With the notation introduced above, \(H^r_\mathrm {cts}(K_\mathrm {tame}, {\mathbb {Z}_l})=0\) for \(r=1,2,3\).

### Proof

*r*we have

*U*runs through the finite sets of tame primes. For such primes

*p*we have \(H^r_\mathrm {cts}(K_p,\mathbb {Q}_l/{\mathbb {Z}_l})=0\) for \(r=1,2\). Hence by an obvious long exact sequence we have \(H^r_\mathrm {cts}(K_p,\mathbb {Z}/l^t)=0\) for \(r=1,2\). By the Künneth formula we have

*p*. Since \(G(\mathbb {F}_p)\) is finite, we have \(H^3(G(\mathbb {F}_p),\mathbb {Q}_l)=0\), and therefore \(H^3(G(\mathbb {F}_p),{\mathbb {Z}_l})=H^2(G(\mathbb {F}_p),\mathbb {Q}_l/{\mathbb {Z}_l})=0\). From this it follows that

(It might be tempting to imagine that the result above can be extended further in a simple way. However, we note that the projective system in Eq. 7 does not satisfy the Mittag–Leffler condition, so we do not expect \(H^4_\mathrm {cts}(K_\mathrm {tame},{\mathbb {Z}_l})\) to be finitely generated as a \({\mathbb {Z}_l}\)-module).

### Lemma 5

*L*be an

*S*-arithmetic level in \({\widehat{G(\mathbb {Q})}}\). For \(r=0,1,2,3\) we have

*G*over \(\mathbb {Q}\).

### Proof

*p*is a prime in

*W*, which is not equal to

*l*. The group \(K_p\) contains a normal pro-

*p*subgroup of finite index. From this it follows that

*p*is a prime in

*S*which is not equal to

*l*. We recall from [7] that there is a spectral sequence which calculates the cohomology of \(G(\mathbb {Q}_p)\) in terms of the cohomology of its compact open subgroups. Let \(K_{p}^0\) be a maximal pro-

*p*subgroup of \(G(\mathbb {Q}_p)\). The subgroup \(K_{p}^0\) is compact and open in \(G(\mathbb {Q}_p)\). There are finitely many maximal compact subgroups of \(G(\mathbb {Q}_p)\), which contain \(K_p^0\); we call these subgroups \(K_{1},\ldots ,K_{n}\). In the spectral sequence, the \(E_{1}\)-sheet is given by

*n*vertices. As this simplex is contractable, we have

*l*is in

*S*or not. In either case we have \(H^\bullet _\mathrm {cts}(L_l,\mathbb {Q}_l)= H^\bullet (\mathfrak {g}\otimes \mathbb {Q}_l,\mathbb {Q}_l)\); this is proved in [14] for \(K_l\) and in [7] for \(G(\mathbb {Q}_l)\). As a result of this we have \(H^r_\mathrm {cts}(L,\mathbb {Q}_l)=H^r(\mathfrak {g}\otimes \mathbb {Q}_l,\mathbb {Q}_l)\) for \(r \le 3\). \(\square \)

### Lemma 6

We have \(H^0(\mathfrak {g}\otimes \mathbb {Q}_l,\mathbb {Q}_l)=\mathbb {Q}_l\), \(H^1(\mathfrak {g}\otimes \mathbb {Q}_l,\mathbb {Q}_l)=0\), \(H^2(\mathfrak {g}\otimes \mathbb {Q}_l,\mathbb {Q}_l)=0\) and \(H^3(\mathfrak {g}\otimes \mathbb {Q}_l,\mathbb {Q}_l)=\mathbb {Q}_l^b\), where *b* is the number of simple components of \(G \times _\mathbb {Q}\mathbb {C}\). (Note that in the notation of the introduction we have \(b=b_\mathbb {R}+2b_\mathbb {C}\)).

### Proof

### 6.3 The end of the proof

*S*of the sequences in Eq. 8. Since the sequences consist of finite dimensional vector spaces, the derived functors \( \left( \begin{array}{c} \lim \\ {\mathop {\scriptstyle S}\limits ^{\textstyle \leftarrow }} \end{array}\right) ^{1} \) all vanish, so we have the following exact sequence:

*S*) of the groups \(H^r({\varGamma (L)},\mathbb {Q}_l)\) is \(H^r(G(\mathbb {Q}),\mathbb {Q}_l)\).

The dimensions of the groups \(H^r(G(\mathbb {Q}),\mathbb {Q}_l)\) are the same as those of \(H^r(G(\mathbb {Q}),\mathbb {C})\), and by Proposition 2 these are the same as the the relative Lie algebra cohomology groups \(H^r(\mathfrak {g},\mathfrak {k},\mathbb {C})\). Here \(\mathfrak {k}\) is the Lie algebra of a maximal compact subgroup \(K_\infty \) of \(G(\mathbb {R})\).

Since \({\bar{H}}^2({\mathbb {Z}_l})\) is torsion-free, it follows that \(\bar{H}^2({\mathbb {Z}_l})^{\widehat{G(\mathbb {Q})}}\) is a torsion-free \({\mathbb {Z}_l}\)-module, which spans \(\bar{H}^2(\mathbb {Q}_l)^{\widehat{G(\mathbb {Q})}}\). On the other hand, \({\bar{H}}^2({\mathbb {Z}_l})\) has no non-zero divisible elements. This implies that \({\bar{H}}^2({\mathbb {Z}_l})^{\widehat{G(\mathbb {Q})}}\cong {\mathbb {Z}_l}^{c}\), where \(c= \dim {\bar{H}}^2(\mathbb {Q}_l)^{\widehat{G(\mathbb {Q})}}\). This finishes the proof of the Theorem 6.

### Corollary

There is a subgroup of \({\bar{H}}^2(\mathbb {Z}/l^t)^{\widehat{G(\mathbb {Q})}}\) isomorphic to \((\mathbb {Z}/l^t)^{c}\), all of whose elements virtually lift to characteristic zero, where \(c=\mathrm {rank}_{{\mathbb {Z}_l}} \left( {\bar{H}}^2({\mathbb {Z}_l})^{\widehat{G(\mathbb {Q})}}\right) \).

### Proof

*A*for the subgroup of elements in \({\bar{H}}^2(\mathbb {Z}/l^t)\) which virtually lift to characteristic zero. By definition,

*A*is the image of \({\bar{H}}^2({\mathbb {Z}_l})\) in \(H^2(\mathbb {Z}/l^t)\). We therefore have a short exact sequence

### Acknowlegements

I’d like to thank Lars Louder for many useful discussions. I’d also like to thank the anonymous referee for suggesting several improvements.

## Footnotes

- 1.
This is a proper subgroup if and only if \(r\ge 3\).

## References

- 1.Bestvina, M.: Questions in Geometric Group Theory. http://www.math.utah.edu/~bestvina (Updated July 2004)
- 2.Blasius, D., Franke, J., Grünewald, F.: Cohomology of \(S\)-arithmetic groups in the number field case. Invent. Math.
**116**, 75–93 (1994)MathSciNetCrossRefGoogle Scholar - 3.Borel, A.: Sur la cohomology des espaces fibres principaux et des espaces homogenes de groupes de lie compacts. Ann. Math.
**57**(1), 115–207 (1953)MathSciNetCrossRefGoogle Scholar - 4.Borel, A., Serre, J.-P.: Corners and Arithmetic groups. Comment. Math. Helv.
**48**, 436–491 (1973)MathSciNetCrossRefGoogle Scholar - 5.Borel, A., Wallach, N.R.: Continuous cohomology, discrete subgroups, and representations of reductive groups. In: Sard, A. (ed.) Mathematical Surveys and Monographs, vol. 67, 2nd edn. American Mathematical Society, Providence, RI (2000)Google Scholar
- 6.Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, Princeton (1956)zbMATHGoogle Scholar
- 7.Casselman, W., Wigner, D.: Continuous cohomology and a conjecture of Serre’s. Invent. Math.
**25**, 199–211 (1974)MathSciNetCrossRefGoogle Scholar - 8.Flach, M.: Cohomology of topological groups with applications to the Weil group. Compos. Math.
**144**(3), 633–656 (2008)MathSciNetCrossRefGoogle Scholar - 9.Gromov, M.: Word hyperbolic groups. In: Gersten, S.M. (ed.) Essays in Group Theory. Mathematical Sciences Research Institute Publications, vol. 8, pp. 75–264. Springer, New York (1987)Google Scholar
- 10.Gromov, M.: Geometric group theory. In: Niblo, G.A., Roller, M.A. (eds.) Asymptotic Invariants of Infinite Groups. LMS Lecture Note Series 182, vol. 2. Cambridge University Press, Cambridge (1993)Google Scholar
- 11.Kapovich, I., Wise, D.T.: The equivalence of some residual properties of word-hyperbolic groups. J. Algebr.
**223**(2), 562–583 (2000)MathSciNetCrossRefGoogle Scholar - 12.Kneser, M.: Normalteiler ganzzahliger Spingruppen. J. Reine Angew. Math.
**311**(312), 191–214 (1979)MathSciNetzbMATHGoogle Scholar - 13.Kneser, M.: Strong approximation. In: Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), pp. 187–196. American Mathematical Society, Providence, RI (1966)Google Scholar
- 14.Lazard, M.: Groupes analytiques \(p\)-adiques. Publ. Math. l’IHÉS
**26**, 5–219 (1965)CrossRefGoogle Scholar - 15.Lubotzki, A., Manning, J. F., Wilton, H.: Generalized triangle groups, expanders, and a problem of Agol and Wise.
*preprint 2018*(arXiv:1702.08200) (2018) - 16.Moore, C.C.: Extensions and low dimensional cohomology theory of locally compact groups I. Trans. Am.Math. Soc.
**113**, 40–63 (1964)MathSciNetzbMATHGoogle Scholar - 17.Moore, C.C.: Extensions and low dimensional cohomology theory of locally compact groups II. Trans. Am.Math. Soc.
**113**, 64–86 (1964)MathSciNetzbMATHGoogle Scholar - 18.Moore, C.C.: Group extensions and cohomology for locally compact groups. III. Trans. Am.Math. Soc.
**221**(1), 1–33 (1976)MathSciNetCrossRefGoogle Scholar - 19.Quillen, D.: On the cohomology and K-theory of the general linear groups over a finite field. Ann. Math. Second Ser.
**96**(3), 552–586 (1972)MathSciNetCrossRefGoogle Scholar - 20.Serre, J.-P.: Le probleme des groupes de congruence pour \({{\rm SL}}_2\). Ann. Math.
**92**(3), 489–527 (1970)MathSciNetCrossRefGoogle Scholar - 21.Serre, J.-P.: Cohomologie Galoisienne. Lecture Notes in Mathematics, vol. 5. Springer, Berlin (1994)CrossRefGoogle Scholar
- 22.Steinberg, R.: Lectures on Chevalley Groups. Yale University, Yale (1967)zbMATHGoogle Scholar
- 23.Tits, R.: Reductive Groups over Local Fields. In: Automorphic forms, representations and \(L\)-functions, Part 1, Proceedings of Symposium on Pure Mathematics, vol. XXXIII, pp. 29–69 (1979)Google Scholar
- 24.Weibel, C.A.: An Introduction to Homological Algebra. Cambridge Studies in Advanced Mathematics, vol. 38. Cambridge University Press, Cambridge (1994)CrossRefGoogle Scholar
- 25.Wigner, D.: Algebraic cohomology of topological groups. Trans. Am. Math. Soc.
**178**, 83–93 (1973)MathSciNetCrossRefGoogle Scholar - 26.Wise, D.T.: The residual finiteness of negatively curved polygons of finite groups. Invent. Math.
**149**(3), 579–617 (2002)MathSciNetCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.