S-arithmetic groups of SL_2 type

Arithmetic Groups and the Lehmer Conjecture

We generalize a result of Sury and prove that uniform discreteness of cocompact lattices in higher rank semisimple Lie groups (first conjectured by Margulis) is equivalent to a weak form of Lehmer's conjecture. We include a short survey of related results and conjectures.

Borel's stable range for the cohomology of arithmetic groups

In this note, we remark on the range in Borel's theorem on the stable cohomology of the arithmetic groups Sp(2n,Z) and SO(n,n;Z). This improves the range stated in Borel's original papers, an improvement that was known to Borel. Our main task is a technical computation involving the Weyl group action on roots and weights. This note originally appeared as the appendix to arXiv:1711.03139.

关于S-算术群的欧拉特征

On the Euler characteristic of $S$-arithmetic groups

该研究证明了S-算术群的欧拉特征符号仅依赖于S-同余完成，扩展了之前的相关研究成果。

Arithmetic Sets in Groups

We define a notion of an arithmetic set in an arbitrary countable group and study properties of these sets in the cases of Abelian groups and non-abelian free groups.

Weakly multiplicative arithmetic functions and the normal growth of groups

We show that an arithmetic function which satisfies some weak multiplicativity properties and in addition has a non-decreasing or $\log$-uniformly continuous normal order is close to a function of the form $n\mapsto n^c$. As an application we show that a finitely generated, residually finite, infinite group, whose normal growth has a non-decreasing or a $\log$-uniformly continuous normal order is isomorphic to $(\mathbb{Z}, +)$.

算术群

Arithmetic groups

本文是EPFL 2014年算术格入门课程的讲义，系统介绍算术群的基本概念与理论。

Weakly commensurable S-arithmetic subgroups in almost simple algebraic groups of types B and C

Let G and G' be absolutely almost simple algebraic groups of types B and C respectively, of rank at least 3, and defined over a number field K. We determine when G and G' have the same isomorphism or isogeny classes of maximal K-tori. This leads to the necessary and sufficient conditions for two Zariski-dense S-arithmetic subgroups of G and G' to be weakly commensurable.

有限群的算术性质与算术刻画

On arithmetical properties and arithmetical characterizations of finite groups

本文综述了有限群算术性质及其算术参数刻画的最新结果，基于作者在2022年女性数学世界会议上的报告。

Bounded clustering property characterizes arithmetic nonuniform Kleinian groups

In this short note, we show that the B-C property characterizes arithmetic lattices along all nonuniform lattices in ${\mathrm{PSL}}(2,\mathbb{C})$.

半算术Fuchsian群的几何与算术

Geometry and arithmetic of semi-arithmetic Fuchsian groups

本文引入了半算术Fuchsian群的新几何不变量stretch，基于Riemannian质心概念，并利用算术Margulis引理证明了在算术维数、stretch和余面积有界时仅有有限多个共轭类，无界时则存在无限序列。

Unipotent Generators for Arithmetic Groups

We sketch a simplification of proofs of old results on the arithmeticity of the group generated by opposing integral unipotent radicals in higher rank arithmetic groups

Commensurated subgroups of arithmetic groups, totally disconnected groups and adelic rigidity

Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.

Higher rank arithmetic groups which are not invariably generated

It was conjectured in [KLS14] that for arithmetic groups, Invariable Generation is equivalent to the Congruence Subgroup Property. In view of the famous Serre conjecture this would imply that higher rank arithmetic groups are invariably generated. In this paper we prove that some higher rank arithmetic groups are not invariably generated.

Thin right-angled Coxeter groups in some uniform arithmetic lattices

Using a variant of an unpublished argument due to Agol, we show that an irreducible right-angled Coxeter group on ${n \geq 3}$ vertices embeds as a thin subgroup of a uniform arithmetic lattice in an indefinite orthogonal group $\mathrm{O}(p, q)$ for some $p, q \geq 1$ satisfying $p+q=n$.

Arithmetic graphs of finite groups

In this paper we introduced an arithmetic graph function which associates with every group G the directed graph whose vertices corresponds to the divisors of |G|. With the help of such functions we introduced arithmetic graphs of classes of groups, in particular of hereditary saturated formations. We formulated the problem of the recognition of classes of groups by arithmetic graph functions and investigated this problem for some arithmetic graph functions.

On Rigidity of $S$-Arithmetic Kac-Moody Groups

In this article we investigate rigidity properties of $S$-arithmetic Kac-Moody groups in characteristic $0$.

Commensurators of solvable S-arithmetic groups

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.

Abelianization conjectures for some arithmetic square complex groups

We extend a conjecture of Kimberley-Robertson on the abelianizations of certain square complex groups.

Arithmeticity of Four Hypergeometric Monodromy Groups associated to Calabi-Yau threefolds

In [12], we show that 3 of the 14 hypergeometric monodromy groups associated to Calabi-Yau threefolds, are arithmetic. Brav-Thomas (in [3]) show that 7 of the remaining 11, are thin. In this article, we settle the arithmeticity problem for the 14 monodromy groups, by showing that, the remaining 4 are arithmetic.

S-arithmetic spinor groups with the same finite quotients and distinct $\ell^2$-cohomology

In this note we refine examples by Aka from arithmetic to S-arithmetic groups to show that the vanishing of the $i$-th $\ell^2$-Betti number is not a profinite invariant for all $i \ge 2$.

Quantifying residual finiteness of arithmetic groups

The normal Farb growth of a group quantifies how well-approximated the group is by its finite quotients. We show that any S-arithmetic subgroup of a higher rank Chevalley group G has normal Farb growth n^dim(G).

A new proof of finiteness of maximal arithmetic reflection groups

We give a new proof of the finiteness of maximal arithmetic reflection groups. Our proof is novel in that it makes no use of trace formulas or other tools from the theory of automorphic forms and instead relies on the arithmetic Margulis lemma of Fraczyk, Hurtado and Raimbault.

Arithmetic progressions in finite groups

In this paper, we describe the structure of finite groups whose element orders or proper (abelian) subgroup orders form an arithmetic progression of ratio $r\geq 2$. This extends the case $r=1$ studied in previous papers \cite{1,8,4}.

On the bounded cohomology of semi-simple groups, S-arithmetic groups and products

We prove vanishing results for Lie groups and algebraic groups (over any local field) in bounded cohomology. The main result is a vanishing below twice the rank for semi-simple groups. Related rigidity results are established for S-arithmetic groups and groups over global fields. We also establish vanishing and cohomological rigidity results for products of general locally compact groups and their lattices.

An infinitely generated virtual cohomology group for noncocompact arithmetic groups over function fields

Let G be a noncocompact irreducible arithmetic group over a global function field K of characteristic p, and let H be a finite-index, residually p-finite subgroup of G. We show that the cohomology of H in the dimension of its associated Euclidean building with coefficients in the field of p elements is infinite.

An introduction to arithmetic groups

Arithmetic groups are groups of matrices with integral entries. We shall first discuss their origin in number theory (Gauss, Minkowski) and their role in the "reduction theory of quadratic forms". Then we shall describe these groups by generators and relations. The next topic will be: are all subgroups of finite index given by congruence conditions? Finally, we shall discuss rigidity properties of arithmetic groups.

Arithmetic groups with isomorphic finite quotients

Two finitely generated groups have the same set of finite quotients if and only if their profinite completions are isomorphic. Consider the map which sends (the isomorphism class of) an S-arithmetic group to (the isomorphism class of) its profinite completion. We show that for a wide class of S-arithmetic groups, this map is finite to one, while the the fibers are of unbounded size.

Exponential higher dimensional isoperimetric inequalities for some arithmetic groups

We show that arithmetic subgroups of semisimple groups of relative Q-type A_n, B_n, C_n, D_n, E_6, or E_7 have an exponential lower bound to their isoperimetric inequality in the dimension that is 1 less than the real rank of the semisimple group.

Some more Non-arithmetic Rigid groups

In "Non arithmetic super rigid groups: counter examples to Platonov's conjecture" Bass and Lubotzky gave a counter example to Platonov's conjecture by presenting an example of a linear group with super-rigidity which is not an arithmetic lattice. In this note, a much richer class of such groups is presented with a somewhat simpler proof.

An arithmetic method of counting the subgroups of a finite abelian group

The main goal of this paper is to apply the arithmetic method developed in our previous paper \cite{13} to determine the number of some types of subgroups of finite abelian groups.

Generators for Arithmetic Groups

We prove that any non-cocompact irreducible lattice in a higher rank semi-simple Lie group contains a subgroup of finite index, which has three generators.

Finiteness of arithmetic hyperbolic reflection groups

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

Zariski Density and Computing in Arithmetic Groups

For $n > 2$, let $Γ$ denote either $SL(n, Z)$ or $Sp(n, Z)$. We give a practical algorithm to compute the level of the maximal principal congruence subgroup in an arithmetic group $H\leq Γ$. This forms the main component of our methods for computing with such arithmetic groups $H$. More generally, we provide algorithms for computing with Zariski dense groups in $Γ$. We use our GAP implementation of the algorithms to solve problems that have emerged recently for important classes of linear groups.

Profinite non-rigidity of arithmetic groups

We show that for a typical high rank arithmetic lattice $Γ$, there exist finite index subgroups $Γ_{1}$ and $Γ_{2}$ such that $Γ_{1} \not\simeq Γ_{2}$ while $\widehat{Γ_{1}} \simeq \widehat{Γ_{2}}$. But there are exceptions to that rule.

有限单群及相关群的算术不变量

Arithmetic invariants for finite simple and related groups

该短文研究通过算术不变量刻画有限单群及相关群的问题，提出统一研究思路，梳理领域进展并指出待解决问题。

On the model theory of higher rank arithmetic groups

Let $Γ$ be a centerless irreducible higher rank arithmetic lattice in characteristic zero. We prove that if $Γ$ is either non-uniform or is uniform of orthogonal type and dimension at least 9, then $Γ$ is bi-interpretable with the ring $\mathbb{Z}$ of integers. It follows that the first order theory of $Γ$ is undecidable, that all finitely generated subgroups of $Γ$ are definable, and that $Γ$ is characterized by a single first order sentence among all finitely generated groups.

Automorphism groups of polycyclic-by-finite groups and arithmetic groups

We show that the outer automorphism group of a polycyclic-by-finite group is an arithmetic group. This result follows from a detailed structural analysis of the automorphism groups of such groups. We use an extended version of the theory of the algebraic hull functor initiated by Mostow. We thus make applicable refined methods from the theory of algebraic and arithmetic groups. We also construct examples of polycyclic-by-finite groups which have an automorphism group which does not contain an arithmetic group of finite index. Finally we discuss applications of our results to the groups of homo…

Groups whose same-order types are arithmetic progressions

The same-order type $τ_e(G)$ of a finite group $G$ is a set formed of the sizes of the equivalence classes containing the same order elements of $G$. In this paper, we study an arithmetical property of this set. More exactly, we outline some results on the classification and existence of finite groups whose same-order types are arithmetic progressions formed of 3 or 4 elements, the latter being the maximum size of such a sequence.

Arithmetic Aspects of Bianchi Groups

We discuss several arithmetic aspects of Bianchi groups, especially from a computational point of view. In particular, we consider computing the homology of Bianchi groups together with the Hecke action, connections with automorphic forms, abelian varieties, Galois representations and the torsion in the homology of Bianchi groups. Along the way, we list several open problems and conjectures, survey the related literature, presenting concrete examples and numerical data.

Arithmetical Properties of Finite Groups

Let $G$ be a finite group and $Ch_i(G)$ some quantitative sets. In this paper we study the influence of $Ch_i(G)$ to the structure of $G$. We present a survey of author and his colleagues' recent works.

Infinitely many quasi-arithmetic maximal reflection groups

In contrast to the fact that there are only finitely many maximal arithmetic reflection groups acting on the hyperbolic space $\mathbb{H}^n$, $n\geq 2$, we show that: (a) one can produce infinitely many maximal quasi-arithmetic reflection groups acting on $\mathbb{H}^2$; (b) they admit infinitely many different fields of definition; (c) the degrees of their fields of definition are unbounded. However, for $n\geq 14$ an approach initially developed by Vinberg shows that there are still finitely many fields of definitions in the quasi-arithmetic case.

Arithmeticity of Some Hypergeometric Monodromy Groups in Sp(4)

The article [14] gives a list of 51 symplectic hypergeometric monodromy groups corresponding to primitive pairs of degree four polynomials, which are products of cyclotomic polynomials, and for which, the absolute value of the leading coefficient of the difference polynomial is greater than 2. It follows from [12] and [14] that 12 of the 51 monodromy groups are arithmetic (cf. Table 1); and the thinness of 13 of the remaining 39 monodromy groups follows from [3] (cf. Table 2). In this article, we show that 15 of the remaining 26 monodromy groups are arithmetic (cf. Table 3).

Interpreting the arithmetic in Thompson's group F

We prove that the elementary theory of Thompson's group $F$ is hereditarily undecidable.

Filling boundaries of coarse manifolds in semisimple and solvable arithmetic groups

We provide partial results towards a conjectural generalization of a theorem of Lubotzky-Mozes-Raghunathan for arithmetic groups (over number fields or function fields) that implies, in low dimensions, both polynomial isoperimetric inequalities and finiteness properties. As a tool in our proof, we establish polynomial isoperimetric inequalities and finiteness properties for certain solvable groups that appear as subgroups of parabolic groups in semisimple groups, thus generalizing a theorem of Bux. We also develop a precise version of reduction theory for arithmetic groups whose proof is, for …

Arithmetic Groups (Banff, Alberta, April 14-19, 2013)

We present detailed summaries of the talks that were given during a week-long workshop on Arithmetic Groups at the Banff International Research Station in April 2013. The vast majority of these reports are based on abstracts that were kindly provided by the speakers. Video recordings of many of the lectures are available online.

On the structure of finite groups determined by the arithmetic and geometric means of element orders

In this paper we consider two functions related to the arithmetic and geometric means of element orders of a finite group, showing that certain lower bounds on such functions strongly affect the group structure. In particular, for every prime $p$, we prove a sufficient condition for a finite group to be $p$-nilpotent, that is, a group whose elements of $p'$-order form a normal subgroup. Moreover, we characterize finite cyclic groups with prescribed number of prime divisors.

Some applications of arithmetic groups in cryptography

In this paper we will give various examples of exponentially distorted subgroups in linear groups, including some new example of subgroups of $SL_n(\mathbb{Z}[x])$ for $n \ge 3$, and show how they can be used to construct symmetric-key cryptographic platforms.

Arithmetic aspects of self-similar groups

We prove that an irreducible lattice in a semisimple algebraic group is virtually isomorphic to an arithmetic lattice if and only if it admits a faithful self-similar action on a rooted tree of finite valency.

On automorphisms of arithmetic subgroups of unipotent groups in positive characteristic

Let F be a local field of positive characteristic, and let G be either a Heisenberg group over F, or a certain (nonabelian) two-dimensional unipotent group over F. If H is an arithmetic subgroup of G, we provide an explicit description of every automorphism of H. From this description, it follows that every automorphism of H virtually extends to a virtual automorphism of G.

NonLERFness of arithmetic hyperbolic manifold groups and mixed 3-manifold groups

We will show that, for any noncompact arithmetic hyperbolic $m$-manifold with $m> 3$, and any compact arithmetic hyperbolic $m$-manifold with $m> 4$ that is not a $7$-dimensional arithmetic hyperbolic manifold defined by octonions, its fundamental group is not LERF. The main ingredient in the proof is a study on abelian amalgamations of hyperbolic $3$-manifold groups. We will also show that a compact orientable irreducible $3$-manifold with empty or tori boundary supports a geometric structure if and only if its fundamental group is LERF.

Finiteness properties of arithmetic groups over function fields

We determine when an arithmetic subgroup of a reductive group defined over a global function field is of type FP_\infty by comparing its large-scale geometry to the large-scale geometry of lattices in real semisimple Lie groups.

群扩张中的算术

Arithmetic in Group Extensions

本文提出一种适用于具有非平凡可解根的群的 pc 表示推广，在内存和算术上更高效，并用于构造怪物群的极大子群及计算其未知特征标表。

Finiteness Properties of S-Arithmetic Groups - a Survey

We give an overview about finiteness properties of soluble S-arithmetic groups. Both, the number field case and the function field case are covered. The main result is: If B is a Borel subgroup in a Chevalley group and R is an S-arithmetic ring, then the group B(R) has finiteness length |S|-1 in the function field case, and infinite finiteness length in the number field case.

Arithmetic Groups Have Rational Representation Growth

Let G be an arithmetic lattice in a semisimple algebraic group over a number field. We show that if G has the congruence subgroup property, then the number of n-dimensional irreducible representations of G grows like n^a, where a is a rational number.