Profinite commensurability of $S$-arithmetic groups

In [1], Borel discussed discrete arithmetic groups arising from quaternion algebras over number fields with particular reference to arithmetic Kleinian and arithmetic Fuchsian groups. In these cases, he described, in each commensurability class, a class of groups which contains all maximal groups. Developing results on embedding commutative orders of the defining number field into maximal or Eichler orders in the defining quaternion algebra, Chinburg and Friedman [2] stated necessary and sufficient conditions for the existence of torsion in this class of groups in terms of the defining arithmetic data. This was more fully explored in the case of Kleinian groups in [3]. In the case of Fuchsian groups, these results on the existence of torsion were extended to obtain formulas for the number of conjugacy classes of finite cyclic subgroups for each group in this class [8, 9]. In this paper, we examine, across the range of arithmetic Fuchsian groups, how widespread torsion is in maximal Fuchsian groups. Some studies in low genus cases (see e.g. [7, 12]) indicate that 2-torsion is very prevalent. The results obtained here substantiate that but we will also obtain maximal arithmetic Fuchsian groups which are torsion-free. The author is grateful to Alan Reid for conversations on parts of this paper.

CONTENTS Introduction § 1. Arithmetic groups § 2. Adèle groups § 3. Tamagawa numbers § 4. Approximations in algebraic groups § 5. Class numbers and class groups of algebraic groups § 6. The genus problem in arithmetic groups § 7. Classification of maximal arithmetic subgroups § 8. The congruence problem § 9. Groups of rational points over global fields § 10. Galois cohomology and the Hasse principle § 11. Cohomology of arithmetic groups References

Suppose is a simple and simply connected algebraic group over an algebraic number field and is a finite set of valuations of containing all Archimedean valuations. This paper is a study of the connections between abstract properties of the -arithmetic subgroup and the congruence property, i.e. the finiteness of the corresponding congruence kernel . In particular, it is shown that if the profinite completion satisfies condition (PG), (i.e., for any integer > 0 and any prime there exist and such that for all ), then is finite. Examples are given demonstrating the possibility of effectively verifying (PG)' .

Let X X be a smooth curve over a finite field of characteristic p p , let E E be a number field, and let L = { L λ } \mathbf {L} = \{\mathcal {L}_\lambda \} be an E E -compatible system of lisse sheaves on the curve X X . For each place λ \lambda of E E not lying over p p , the λ \lambda -component of the system L \mathbf {L} is a lisse E λ E_\lambda -sheaf L λ \mathcal {L}_\lambda on X X , whose associated arithmetic monodromy group is an algebraic group over the local field E λ E_\lambda . We use Serre’s theory of Frobenius tori and Lafforgue’s proof of Deligne’s conjecture to show that when the E E -compatible system L \mathbf {L} is semisimple and pure of some integer weight, the isomorphism type of the identity component of these monodromy groups is “independent of λ \lambda ”. More precisely, after replacing E E by a finite extension, there exists a connected split reductive algebraic group G 0 G_0 over the number field E E such that for every place λ \lambda of E E not lying over p p , the identity component of the arithmetic monodromy group of L λ \mathcal {L}_\lambda is isomorphic to the group G 0 G_0 with coefficients extended to the local field E λ E_\lambda .

In this paper we prove local–global principles for the existence of an embedding (E, σ) ↪ (A, τ) of a given global field E endowed with an involutive automorphism σ into a simple algebra A given with an involution τ in all situations except where A is a matrix algebra of even degree over a quaternion division algebra and τ is orthogonal (Theorem A of the introduction). Rather surprisingly, in the latter case we have a result which in some sense is opposite to the local–global principle, viz. algebras with involution locally isomorphic to (A, τ) are distinguished by their maximal subfields invariant under the involution (Theorem B of the introduction). These results can be used in the study of classical groups over global fields. In particular, we use Theorem B to complete the analysis of weakly commensurable Zariski-dense S -arithmetic groups in all absolutely simple algebraic groups of type different from D_4 which was initiated in our paper [23]. More precisely, we prove that in a group of type D_n , n even > 4 , two weakly commensurable Zariski-dense S -arithmetic subgroups are actually commensurable. As indicated in [23], this fact leads to results about length-commensurable and isospectral compact arithmetic hyperbolic manifolds of dimension 4n + 7 , with n ≥ 1 . The appendix contains a Galois-cohomological interpretation of our embedding theorems.

Some finitely presented nilpotent-by-abelian groups

We prove that the number of unimodular integral n × n matrices in a norm ball whose characteristic polynomial has Galois group different than the full symmetric group Sn is of strictly lower order of magnitude than the number of all such matrices in the ball, as the radius increases. More generally, we prove a similar result for the Galois groups associated with elements in any connected semisimple linear algebraic group defined and simple over a number field F . Our method is based on the abstract large sieve method developed by Kowalski, and the study of Galois groups via reductions modulo primes developed by Jouve, Kowalski and Zywina. The two key ingredients are a uniform quantitative lattice point counting result, and a non-concentration phenomenon for lattice points in algebraic subvarieties of the group variety, both established previously by the authors. The results answer a question posed by Rivin and by Jouve, Kowalski and Zywina, who have considered Galois groups of random products of elements in algebraic groups.

Let G be a reductive algebraic group over Q and Γ⊂G(Q) an arithmetic subgroup. Let K∞⊂G(R) be a maximal compact subgroup. We study the asymptotic behavior of the counting functions of the cuspidal and residual spectrum, respectively, of the regular representation of G(R) in L2(Γ﹨G(R)) of a fixed K∞-type σ. A conjecture, which is due to Sarnak, states that the counting function of the cuspidal spectrum of type σ satisfies Weyl's law and the residual spectrum is of lower order growth. Using the Arthur trace formula we reduce the conjecture to a problem about L-functions occurring in the constant terms of Eisenstein series. If G satisfies property (L), introduced by Finis and Lapid, we establish the conjecture. This includes classical groups over a number field.

We classify all one-class genera of admissible lattice chains of length at least 2 in hermitian spaces over number fields. If L is a lattice in the chain and \(\mathfrak {p} \) the prime ideal dividing the index of the lattices in the chain, then the \(\{ \mathfrak {p} \} \)-arithmetic group \(\mathrm {Aut} (L_{ \{ \mathfrak {p} \} }) \) acts chamber transitively on the corresponding Bruhat-Tits building. So our classification provides a step forward to a complete classification of these chamber transitive groups which has been announced 1987 (without a detailed proof) by Kantor, Liebler and Tits. In fact we find all their groups over number fields and one additional building with a discrete chamber transitive group.

We show that the abstract commensurator of an S-arithmetic subgroup of a solvable algebraic group over Q is isomorphic to the Q-points of an algebraic group, and compare this with examples of nonlinear abstract commensurators of S-arithmetic groups in positive characteristic. In particular, we include a description of the abstract commensurator of the lamplighter group.

The work of Nikulin and Agol, Belolipetsky, Storm, and Whyte shows that only finitely many number fields may serve as the field of definition of an arithmetic hyperbolic reflection group. An important problem posed by Nikulin is to enumerate these fields and their degrees. In this paper we prove that in dimension 2 the degree of these fields is at most 7. More generally we prove that the degree of the field of definition of the quaternion algebra associated to an arithmetic Fuchsian group of genus 0 is at most 7, confirming a conjecture of Long, Maclachlan and Reid. We also obtain upper bounds for the discriminants of these fields of definition, allowing for an enumeration which should be useful for the classification of arithmetic hyperbolic reflection groups.

There are errors in the proof of uniqueness of arithmetic subgroups of the smallest covolume. In this note we correct the proof, obtain certain results which were stated as a conjecture, and we give several remarks on further developments. Mathematics Subject Classification (2000): 11E57 (primary); 22E40 (secondary). 1.1. Let us recall some notation and basic notions. Following [1] we will assume that n is even and n ≥ 4. The group of orientation preserving isometries of hyperbolic n-space is isomorphic to SO(1, n)o, the connected component of the identity of the special orthogonal group of signature (1, n), which can be identified with SO0(1, n), the subgroup of SO(1, n) preserving the upper half space. This group is not Zariski closed in SLn+1 thus in order to construct arithmetically defined subgroups of SO(1, n)o we consider arithmetic subgroups of the orthogonal group SO(1, n) or, more precisely, of groups G = SO( f ) where f is an admissible quadratic form defined over a totally real number field k (see [1, Section 2.1]). We have an exact sequence of k-isogenies: 1 → C → G φ → G → 1, (1.1) where G(k) Spin( f ) is the simply connected cover of G and C μ2 is the center of G. This induces an exact sequence in Galois cohomology (see [5, Section 2.2.3]) G(k) φ → G(k) δ → H1(k, C) → H1(k, G). (1.2) The main idea of this note is that by using (1.2) certain questions about arithmetic subgroups of G can be reduced to questions about the Galois cohomology group H1(k, C). A coherent collection of parahoric subgroups P = (Pv)v∈V f of G (V f = V f (k) denotes the set of finite places of the field k) defines a principal arithmetic subgroup Received October 16, 2006; accepted in revised form March 12, 2007.

A group [Formula: see text] is called bounded if every conjugation-invariant norm on [Formula: see text] has finite diameter. We introduce various strengthenings of this property and investigate them in several classes of groups including semisimple Lie groups, arithmetic groups and linear algebraic groups. We provide applications to Hamiltonian dynamics.

A generating function for the Euler characteristic of out( F n)

We investigate sections of the arithmetic fundamental groups of hyperbolic curves over function fields. As a consequence we prove that the anabelian section conjecture of Grothendieck holds over all finitely generated fields over \mathbb Q if it holds over all number fields, under the condition of finiteness (of the \mathcal l -primary parts) of certain Shafarevich–Tate groups. We also prove that if the section conjecture holds over all number fields then it holds over all finitely generated fields for curves which are defined over a number field.