Quasi-isometries of rank one S-arithmetic lattices

Quasi-isometric rigidity of higher rank S-arithmetic lattices

We show that S-arithmetic lattices in semisimple Lie groups with no rank one factors are quasi-isometrically rigid.

Dehn functions of higher rank arithmetic groups of type $A_n$ in products of simple Lie 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.

Higher finiteness properties of reductive arithmetic groups in positive characteristic: the rank theorem

We show that the finiteness length of an $S$-arithmetic subgroup $Γ$ in a noncommutative isotropic absolutely almost simple group $G$ over a global function field is one less than the sum of the local ranks of $G$ taken over the places in $S$. This determines the finiteness properties for arithmetic subgroups in isotropic reductive groups, confirming the conjectured finiteness properties for this class of groups. Our main tool is Behr-Harder reduction theory which we recast in terms of the metric structure of euclidean buildings.

Asymptotic Geometry in the product of Hadamard spaces with rank one isometries

In this article we study asymptotic properties of certain discrete groups $Γ$ acting by isometries on a product $\XX=\XX_1\times \XX_2$ of locally compact Hadamard spaces. The motivation comes from the fact that Kac-Moody groups over finite fields, which can be seen as generalizations of arithmetic groups over function fields, belong to this class of groups. Hence one may ask whether classical properties of discrete subgroups of higher rank Lie groups as in [MR1437472] and [MR1933790] hold in this context. In the first part of the paper we describe the structure of the geometric limit set of $…

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。

将伪阿诺索夫映射类计数为弱收缩等距映射

Counting pseudo-Anosovs as weakly contracting isometries

本文证明在有限型双曲曲面映射类群及更广的层级双曲群的Cayley图中，伪阿诺索夫映射类或Morse元素是普遍的，并建立了拟等距不变量计数理论。

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.

Horospherical limit points of S-arithmetic groups

Suppose Gamma is an S-arithmetic subgroup of a connected, semisimple algebraic group G over a global field Q (of any characteristic). It is well known that Gamma acts by isometries on a certain CAT(0) metric space X_S that is a Cartesian product of Euclidean buildings and Riemannian symmetric spaces. For a point p on the visual boundary of X_S, we show there exists a horoball based at p that is disjoint from some Gamma-orbit in X_S if and only if p lies on the boundary of a certain type of flat in X_S that we call "Q-good." This generalizes a theorem of G.Avramidi and D.W.Morris that character…

Actions of certain arithmetic groups on Gromov hyperbolic spaces

We study the variety of actions of a fixed (Chevalley) group on arbitrary geodesic, Gromov hyperbolic spaces. In high rank we obtain a complete classification. In rank one, we obtain some partial results and give a conjectural picture.

Centrality of the congruence subgroup kernel of higher rank non-uniform arithmetic groups

We give here a simple proof of the centrality of the congruence subgroup kernel in the higher rank isotropic case.

Proper actions of lattices on contractible manifolds

Every lattice H in a connected semi-simple Lie group G acts properly discontinuously by isometries on the contractible manifold G/K (K a maximal compact subgroup of G). We prove that if H acts on a contractible manifold W and if either 1) the action is properly discontinuous, or 2) W is equipped with a complete Riemannian metric, the action is by isometries and with unbounded orbits, G is simple with finite center and rank >1, then dim W is not less than dim G/K.

First order rigidity of non-uniform higher rank arithmetic lattices

If $Γ$ is an irreducible non-uniform higher-rank characteristic zero arithmetic lattice (for example, $SL_n(\mathbb{Z})$, $n \geq 3$) and $Λ$ is a finitely generated group that is elementarily equivalent to $Γ$, then $Λ$ is isomorphic to $Γ$.

Higher finiteness properties of arithmetic approximate lattices: The Rank Theorem for number fields

We introduce geometric and homological finiteness properties for countable approximate groups via coarse geometry and then study these finiteness properties for S-arithmetic reductive approximate groups. For S-arithmetic approximate groups without infinite places we show that the finiteness length is finite and compute this finiteness length explicitly. In the simple case it is one less than the sum of the local ranks. This extends the Rank Theorem of Bux, Köhl and the second author from positive characteristic to characteristic zero. Our proof is based on a geometric version of their proof, b…

$\R$-trees, dual laminations, and compact systems of partial isometries

Let $\FN$ be a free group of finite rank $N \geq 2$, and let $T$ be an $\R$-tree with a very small, minimal action of $\FN$ with dense orbits. For any basis $\CA$ of $\FN$ there exists a {\em heart} $K_{\CA} \subset \bar T$ (= the metric completion of $T$) which is a compact subtree that has the property that the dynamical system of partial isometries $a_{i} : K_{\CA} \cap a_{i} K_{\CA} \to a_{i}\inv K_{\CA} \cap K_{\CA}$, for each $a_{i} \in \CA$, defines a tree $T_{(K_{\CA}, \CA)}$ which contains an isometric copy of $T$ as minimal subtree.

Cusped spaces and quasi-isometries of relatively hyperbolic groups

A group $Γ$ with a family of subgroups $\mathbb{P}$ is relatively hyperbolic if $Γ$ admits a cusp-uniform action on a proper $δ$--hyperbolic space. We show that any two such spaces for a given group pair are quasi-isometric, provided the spaces have "constant horospherical distortion," a condition satisfied by Groves--Manning's cusped Cayley graph and by all negatively curved symmetric spaces. Consequently the Bowditch boundary admits a canonical quasisymmetric structure, which coincides with the "naturally occurring" quasisymmetric structure of the symmetric space when considering lattices in…

Geometry and Dynamics of Discrete Isometry Groups of Higher Rank Symmetric Spaces

For real hyperbolic spaces, the dynamics of individual isometries and the geometry of the limit set of nonelementary discrete isometry groups have been studied in great detail. Most of the results were generalised to discrete isometry groups of simply connected Riemannian manifolds of pinched negative curvature. For symmetric spaces of higher rank, which contain isometrically embedded Euclidean planes, the situation becomes far more complicated. This paper is devoted to the study of the geometric limit set of ``nonelementary'' discrete isometry groups of higher rank symmetric spaces. We obtain…

Finiteness properties of soluble arithmetic groups over global function fields

Let G be a Chevalley group scheme and B<=G a Borel subgroup scheme, both defined over Z. Let K be a global function field, S be a finite non-empty set of places over K, and O_S be the corresponding S-arithmetic ring. Then, the S-arithmetic group B(O_S) is of type F_{|S|-1} but not of type FP_{|S|}. Moreover one can derive lower and upper bounds for the geometric invariants Σ^m(B(O_S)). These are sharp if G has rank 1. For higher ranks, the estimates imply that normal subgroups of B(O_S) with abelian quotients, generically, satisfy strong finiteness conditions.

Coxeter Groups are not higher rank Arithmetic Groups

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.

Isometry groups of non-positively curved spaces: discrete subgroups

We study lattices in non-positively curved metric spaces. Borel density is established in that setting as well as a form of Mostow rigidity. A converse to the flat torus theorem is provided. Geometric arithmeticity results are obtained after a detour through superrigidity and arithmeticity of abstract lattices. Residual finiteness of lattices is also studied. Riemannian symmetric spaces are characterised amongst CAT(0) spaces admitting lattices in terms of the existence of parabolic isometries.

On the Asymptotic Number of Generators of High Rank Arithmetic Lattices

$ $Abert, Gelander and Nikolov [AGN17] conjectured that the number of generators $d(Γ)$ of a lattice $Γ$ in a high rank simple Lie group $H$ grows sub-linearly with $v = μ(H / Γ)$, the co-volume of $Γ$ in $H$. We prove this for non-uniform lattices in a very strong form, showing that for $2-$generic such $H$'s, $d(Γ) = O_H(\log v / \log \log v)$, which is essentially optimal. While we can not prove a new upper bound for uniform lattices, we will show that for such lattices one can not expect to achieve a better bound than $d(Γ) = O(\log v)$.

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.

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

格点在$S$-算术群中的流形作用

Actions of lattices in $p$-adic and $S$-arithmetic groups on manifolds

本文证明，当维数小于秩时，简单$p$-adic群中格点在紧流形上的$C^1$微分同胚作用是有限的，并将此结果推广到全不连通的$S$-算术群中的格点。

Quaternionic Arithmetic Lattices of Rank 2 and a Fake Quadric in Characteristic 2

We construct a torsion-free arithmetic lattice in $\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))\times\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))$ arising from a quaternion algebra over $\mathbb{F}_2(z)$. It is the fundamental group of a square complex with universal covering $T_3\times T_3$, a product of trees with constant valency $3$, which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over $\mathbb{F}_2(\!(t)\!)$ by means of non-archimedean uniformization.

算术群的Banach表示的固定点性质与上同调

Fixed point properties and cohomology of Banach representations of arithmetic groups

本文研究半单群格的固定点定理，通过证明高秩单群格在无不变向量时秩以下$L^p$上同调消失，推出低维无环复形作用有有限轨道，并证明Farb与Gromov的猜想。

Growth of conjugacy classes of Schottky groups in higher rank symmetric spaces

Let $X$ be a globally symmetric space of noncompact type, and $Γ\subset\Isom(X)$ a Schottky group of axial isometries. Then $M:=X/Γ$ is a locally symmetric Riemannian manifold of infinite volume. The goal of this note is to give an asymptotic estimate for the number of primitive closed geodesics in $M$ modulo free homotopy with period less than $t$.

正交型一致高阶算术群中的共轭宽度

Conjugacy width in uniform higher rank arithmetic groups of orthogonal type

研究正交型各向异性高阶算术群的共轭宽度，在GRH下证明次数≥17且含非阿基米德赋值时宽度有界，并建立宽度界与同余子群问题的等价联系。

Filling functions of arithmetic groups

The Dehn function and its higher-dimensional generalizations measure the difficulty of filling a sphere in a space by a ball. In nonpositively curved spaces, one can construct fillings using geodesics, but fillings become more complicated in subsets of nonpositively curved spaces, such as lattices in symmetric spaces. In this paper, we prove sharp filling inequalities for (arithmetic) lattices in higher rank semisimple Lie groups. When $n$ is less than the rank of the associated symmetric space, we show that the $n$-dimensional filling volume function of the lattice grows at the same rate as t…

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.

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.

Introduction to Arithmetic Groups

This book provides a gentle introduction to the study of arithmetic subgroups of semisimple Lie groups. This means that the goal is to understand the group SL(n,Z) and certain of its subgroups. Among the major results discussed in the later chapters are the Mostow Rigidity Theorem, the Margulis Superrigidity Theorem, Ratner's Theorems, and the classification of arithmetic subgroups of classical groups. As background for the proofs of these theorems, the book provides primers on lattice subgroups, arithmetic groups, real rank and Q-rank, ergodic theory, unitary representations, amenability, Kaz…

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).

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.

高秩算术群的 Charmenability

Charmenability of higher rank arithmetic groups

本文研究了任意特征下高秩单代数群格上特征的存在性与分类，扩展了非交换 Nevo-Zimmer 结构定理，并推导出非交换 Margulis 因子定理，应用于拓扑动力学、遍历理论、酉表示和算子代数。

可解群的自同构群

Automorphism groups of solvable groups of finite abelian ranks

本文研究了可解群的自同构群结构，通过有理代数包络的构造，探讨了S-算术子群的性质及外自同构群的算术像。

解析 torsion 和上同调 torsion 对于有理数秩为 1 的算术群的渐进行为

Asymptotic behaviour of analytic torsion and cohomological torsion for $\mathbb{Q}$-rank $1$ arithmetic groups

本文扩展了先前工作中关于 SL(n) 同余子群的解析 torsion 的精细渐进行为，应用于获得 SL(2, O_F) 同余子群和 SO(n,1) 同余子群的上同调 torsion 增长的新渐进行为和界。

Gromov's measure equivalence and rigidity of higher rank lattices

In this paper the notion of Measure Equivalence (ME) of countable groups is studied. ME was introduced by Gromov as a measure-theoretic analog of quasi-isometries. All lattices in the same locally compact group are Measure Equivalent; this is one of the motivations for this notion. The main result of this paper is ME rigidity of higher rank lattices: any countable group which is ME to a lattice in a simple Lie group G of higher rank, is commensurable to a lattice in G.

An ergodic theorem for the quasi-regular representation of the free group

In \cite{BAMU}, an ergodic theorem à la Birkhoff-von Neumann for the action of the fundamental group of a compact negatively curved manifold on the boundary of its universal cover is proved. A quick corollary is the irreducibility of the associated unitary representation. These results are generalized \cite{BOYER} to the context of convex cocompact groups of isometries of a CAT(-1) space, using Theorem 4.1.1 of \cite{ROBLI}, with the hypothesis of non arithmeticity of the spectrum. We prove all the analog results in the case of the free group $\mathbb{F}_r$ of rank $r$ even if $\mathbb{F}_r$ i…

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.