Christophe Soule
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.
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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
Investigations into and around a 30-year old conjecture of Gregory Margulis and Robert Zimmer on the commensurated subgroups of S-arithmetic groups.
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.
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.
In this article we investigate rigidity properties of $S$-arithmetic Kac-Moody groups in characteristic $0$.
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.
We extend a conjecture of Kimberley-Robertson on the abelianizations of certain square complex groups.
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.
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 $i \ge 2$.
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).
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}.
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.
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.
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.
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.
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.
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.
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.
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.
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.
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。
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.
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).
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 …
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.
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.
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.
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.
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。
Let W be an irreducible finitely generated Coxeter group. The geometric representation of W in GL(V) provides a discrete embedding in the orthogonal group of the Tits form (the associated bilinear form of the Coxeter group). If the Tits form of the Coxeter group is non-positive and non-degenerate, the Coxeter group does not contain any finite index subgroup isomorphic to an irreducible lattice in a semisimple group of R-rank greater or equal to 2.
Let f be a nondegenerate quadratic form in at least 5 variables over a number field K and let S be a finite set of valuations of K containing all Archimedean ones. We prove that if the Witt index of f is at least 2 or it is 1 and S contains a non-Archimedean valuation, then the S-arithmetic subgroups of the special orthogonal group of f have bounded generation. These groups provide a series of examples of boundedly generated S-arithmetic groups in isotropic, but not quasi-split, algebraic groups.
We prove that the sign of the Euler characteristic of arithmetic groups with CSP is determined by the profinite completion. In contrast, we construct examples showing that this is not true for the Euler characteristic itself and that the sign of the Euler characteristic is not profinite among general residually finite groups of type $F$. Our methods imply similar results for $\ell^2$-torsion as well as a strong profiniteness statement for Novikov-Shubin invariants.
In the 1990s, J.H. Conway published a combinatorial-geometric method for analyzing integer-valued binary quadratic forms (BQFs). Using a visualization he named the "topograph," Conway revisited the reduction of BQFs and the solution of quadratic Diophantine equations such as Pell's equation. It appears that the crux of his method is the coincidence between the arithmetic group $PGL_2({\mathbb Z})$ and the Coxeter group of type $(3,\infty)$. There are many arithmetic Coxeter groups, and each may have unforeseen applications to arithmetic. We introduce Conway's topograph, and generalizations to …
We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection 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…
We prove that the elementary theory of Thompson's group $F$ is hereditarily undecidable.
Given an S-arithmetic group, we ask how much information on the ambient algebraic group, number field of definition, and set of places S is encoded in the commensurability class of the profinite completion. As a first step, we show that the profinite commensurability class of a higher rank S-arithmetic group determines the number field up to arithmetical equivalence and the places in S above unramified primes. We include applications to profiniteness questions of group invariants.
This thesis studies arithmetic of linear algebraic groups. It involves studying the properties of linear algebraic groups defined over global fields, local fields and finite fields, or more generally the study of the linear algebraic groups defined over the fields which admit arbitrary cyclic extensions.
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.
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.
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.
We give a sufficient condition on a pair of (primitive) integral polynomials that the associated hypergeometric group (monodromy group of the corresponding hypergeometric differential equation) is an arithmetic subgroup of the integral symplectic group.
For a number field K, we show that any S-arithmetic subgroup of SL_2(K) contains a subgroup of finite index generated by three elements if card(S)> 1.
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.
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.
This paper contains the results of my PhD-thesis. I will show the K- and L-theoretic Farrell-Jones conjecture (FJC) for the general linear groups over the rationals and over the rational functions over a finite field. This especially implies the conjecture for all S-arithmetic groups.
Suppose $Γ$ is an arithmetic group defined over a global field $K$, that the $K$-type of $Γ$ is $A_n$ with $n \geq 2$, and that the ambient semisimple group that contains $Γ$ as a lattice has at least two noncocompact factors. We use results from Bestvina-Eskin-Wortman and Cornulier-Tessera to show that $Γ$ has a polynomially bounded Dehn function.
A Jørgensen group is a non-elementary Kleinian group that can be generated by two elements for which equality holds in Jørgensen's Inequality. This paper shows that the only torsion-free Jørgensen group is the figure-eight knot group, identifies all non-cocompact arithmetic Jørgensen groups, and establishes a characterization of cocompact arithmetic Jørgensen groups. The paper also defines and computes the Jørgensen number of several non-cocompact Kleinian groups including some two-bridge knot and link 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.
We correct a mistake in the paper "On the cuspidal cohomology of $S$-arithmetic subgroups of reductive groups over number fields " by A. Borel, J.-P. Labesse, J. Schwermer that appeared in Compositio Mathematica, 102 (1996), 1-40.
By classifying $S$-maximal amenable subgroups of algebraic groups over a global field of characteristic zero, we obtain a complete classification of maximal amenable subgroups up to commensurability in the respective arithmetic groups. Futhermore, we prove that these commensurably maximal amenable subgroups are singular and therefore give rise to maximal amenable von Neumann subalgebras.
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。
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。
We develop a practical algorithm to decide whether a finitely generated subgroup of a solvable algebraic group $G$ is arithmetic. This incorporates a procedure to compute a generating set of an arithmetic subgroup of $G$. We also provide a simple new algorithm for integrality testing of finitely generated solvable-by-finite linear groups over the rational field. The algorithms have been implemented in {\sc Magma}.
We give here a simple proof of the centrality of the congruence subgroup kernel in the higher rank isotropic case.
This paper have serious error in the proof of main theorem 1.1.Result is not proved.
We compute invariants of quadratic forms associated to orthogonal hypergeometric groups of degree five. This allows us to determine some commensurabilities between these groups, as well as to say when some thin groups cannot be conjugate to each other.
The Arithmetic is interpreted in all the groups of Richard Thompson and Graham Higman, as well as in other groups of piecewise affine permutations of an interval which generalize the groups of Thompson and Higman. In particular, the elementary theories of all these groups are undecidable. Moreover, Thompson's group $F$ and some of its generalizations interpret the Arithmetic without parameters.
Fixing an arithmetic lattice $Γ$ in an algebraic group $G$, the commensurability growth function assigns to each $n$ the cardinality of the set of subgroups $Δ$ with $[Γ: Γ\cap Δ] [Δ: Γ\cap Δ] = n$. This growth function gives a new setting where methods of F. Grunewald, D. Segal, and G. C. Smith's "Subgroups of finite index in nilpotent groups" apply to study arithmetic lattices in an algebraic group. In particular, we show that for any unipotent algebraic $\mathbb{Z}$-group with arithmetic lattice $Γ$, the Dirichlet function associated to the commensurability growth function satisfies an Eule…
The trace set of a Fuchsian group $Γ$ ist the set of length of closed geodesics in the surface $Γ\backslash \mathbb{H}$. Luo and Sarnak showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering property. Sarnak then conjectured that the B-C property actually characterizes arithmetic Fuchsian groups. Schmutz stated the even stronger conjecture that a cofinite Fuchsian group is arithmetic if its trace set has linear growth. He proposed a proof of this conjecture in the case when the group $Γ$ contains at least one parabolic element, but unfortunately this…
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