We establish lower bounds on the dimensions in which arithmetic groups with torsion can act on acyclic manifolds and homology spheres. The bounds rely on the existence of elementary p-groups in the groups concerned. In some cases, including Sp(2n,Z), the bounds we obtain are sharp: if X is a generalized Z/3-homology sphere of dimension less than 2n-1 or a Z/3-acyclic Z/3-homology manifold of dimension less than 2n, and if n \geq 3, then any action of Sp(2n,Z) by homeomorphisms on X is trivial; if n = 2, then every action of Sp(2n,Z) on X factors through the abelianization of Sp(4,Z), which is …

2013-06-13math.GRmath.GT

Poincaré series of Lie lattices and representation zeta functions of arithmetic groups

We compute explicit formulae for Dirichlet generating functions enumerating finite-dimensional irreducible complex representations of potent and saturable principal congruence subgroups of $\mathrm{SL}_4^m(\mathfrak{o})$ ($m\in\mathbb{N}$) for $\mathfrak{o}$ a compact DVR of characteristic $0$ and odd residue field characteristic. In doing so we develop a novel method for computing Poincaré series associated with commutator matrices of $\mathfrak{o}$-Lie lattices with finite abelianization and whose rank-loci enjoy an additional smoothness property. We give explicit formulae for the abscissa o…

2017-08-29math.GR

Abelianization and fixed point properties of units in integral group rings

Let $G$ be a finite group and $\mathcal{U} (\mathbb{Z} G)$ the unit group of the integral group ring $\mathbb{Z} G$. We prove a unit theorem, namely a characterization of when $\mathcal{U}(\mathbb{Z}G)$ satisfies Kazhdan's property $(\operatorname{T})$, both in terms of the finite group $G$ and in terms of the simple components of the semisimple algebra $\mathbb{Q}G$. Furthermore, it is shown that for $\mathcal{U}( \mathbb{Z} G)$ this property is equivalent to the weaker property $\operatorname{FAb}$ (i.e. every subgroup of finite index has finite abelianization), and in particular also to a h…

2021-01-22math.GRmath.RAmath.RT

On the divisibility of $#\Hom(Γ,G)$ by $|G|

We extend and reformulate a result of Solomon on the divisibility of the title. We show, for example, that if $Γ$ is a finitely generated group, then $|G|$ divides $#\Hom(Γ,G)$ for every finite group $G$ if and only if $Γ$ has infinite abelianization. As a consequence we obtain some arithmetic properties of the number of subgroups of a given index in such a group $Γ$.

2011-05-30math.COmath.GR

算术群的玻尔紧化

The Bohr compactification of an arithmetic group

本文研究算术群的玻尔紧化结构，通过阿贝尔化、列维分解及高R-秩单群情形，将Bohr(Γ)分解为紧李因子与投射有限完备化的乘积或半直积形式。

2023-04-18math.GR

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Arithmetic monodromy actions on pro-metabelian fundamental groups of once-punctured elliptic curves

We prove structure theorems for the moduli stack of elliptic curves equipped with $G$-structures, where $G$ is a finite 2-generated metabelian group. In particular, we show that if $G$ has exponent $e$, then there is a subgroup $H\le GL_2(\mathbb{Z}/e)$ such that $G$-structures on elliptic curves $E$ are equivalent to "congruence structures of level $H$". Our methods are almost entirely group theoretic. Let $\widehat{M}$ denote the free profinite metabelian group of rank 2, then along the way we prove a decomposition of $Out(\widehat{M})$ as an internal semi-direct product of the subgroup of "…

2017-10-16math.AGmath.GRmath.NT

Abelian subgroup structure of square complex groups and arithmetic of quaternions

A square complex is a 2-complex formed by gluing squares together. This article is concerned with the fundamental group $Γ$ of certain square complexes of nonpositive curvature, related to quaternion algebras. The abelian subgroup structure of $Γ$ is studied in some detail.

2005-01-09math.GRmath.GT

Isotropic nonarchimedean S-arithmetic groups are not left orderable

Let A = Z[c], where c is an irrational number whose square is rational, or let A = Z[1/r], where r > 1 is a square-free natural number. We show that no finite-index subgroup of SL(2,A) is left orderable. (Equivalently, these subgroups have no nontrivial orientation-preserving actions on the real line.) This implies that if G is an isotropic F-simple algebraic group over an algebraic number field F, then no nonarchimedean S-arithmetic subgroup of G is left orderable. Our proofs are based on the fact, proved by B.Liehl, that every element of SL(2,A) is a product of a bounded number of elementary…

2004-06-03math.GR

Correlations of sums of two squares and other arithmetic functions in function fields

We investigate a function field analogue of a recent conjecture on autocorrelations of sums of two squares by Freiberg, Kurlberg and Rosenzweig, which generalizes an older conjecture by Connors and Keating. In particular, we provide extensive numerical evidence and prove it in the large finite field limit. Our method can also handle correlations of other arithmetic functions and we give applications to (function field analogues of) the average of sums of two squares on shifted primes, and to autocorrelations of higher divisor functions twisted by a quadratic character.

2017-01-15math.ACmath.GRmath.NT

Bottom of the Length Spectrum of Arithmetic Orbifolds

We prove that cocompact arithmetic lattices in a simple Lie group are uniformly discrete if and only if the Salem numbers are uniformly bounded away from $1$. We also prove an analogous result for semisimple Lie groups. Finally, we shed some light on the structure of the bottom of the length spectrum of an arithmetic orbifold $Γ\backslash X$ by showing the existence of a positive constant $δ(X)>0$ such that squares of lengths of closed geodesics shorter than $δ$ must be pairwise linearly dependent over $\mathbb Q$.

2022-07-05math.GRmath.GTmath.NT