Related papers: Asymptotic cones of finitely presented groups

We construct a nonuniform lattice and an infinite family of uniform lattices in the automorphism group of a hyperbolic building with all links a fixed finite building of rank 2 associated to a Chevalley group. We use complexes of groups and…

We consider an inverse problem associated with some 2-dimensional non-compact surfaces with conical singularities, cusps and regular ends. Our motivating example is a Riemann surface $\mathcal M = \Gamma\backslash{\bf H}^2$ associated with…

Let $G$ be a connected reductive group defined over $\mathbb F_q$. Fix an integer $M\geq 2$, and consider the power map $x\mapsto x^M$ on $G$. We denote the image of $G(\mathbb F_q)$ under this map by $G(\mathbb F_q)^M$ and estimate what…

We show that each end of a noncompact self-shrinker in $\mathbb{R}^3$ of finite topology is smoothly asymptotic to either a regular cone or a self-shrinking round cylinder.

An element $g$ of a group $G$ is a test element if every endomorphism of $G$ that fixes $g$ is an automorphism. Let $G$ be a free group of finite rank, an orientable surface group of genus $n \geq 2$, or a non-orientable surface group of…

We prove some general asymptotic formula for the values of $L$-function of a sequence of constructible $\mathbb Q_l$-sheaves on curves over $\mathbb F_q$ with some good asymptotic properties. We also give the asymptotic formula for the…

Asymptotic cones of metric spaces were first invented by Gromov. They are metric spaces which capture the 'large-scale structure' of the underlying metric space. Later, van den Dries and Wilkie gave a more general construction of asymptotic…

We prove the existence of surface subgroups within any cocompact lattice $\Gamma$ in $\mathrm{SO}(2n,1)$ for $n\geq2$. This result addresses the cases missing from the work of Hamenst\"adt in 2015, who constructed surface subgroups in…

We prove that for a finitely generated linear group G over a field of positive characteristic the family of quotients by finite subgroups has finite asymptotic dimension. We use this to show that the K-theoretic assembly map for the family…

We enumerate rooted 2-connected and 3-connected surface maps with respect to vertices and edges. We also derive the bivariate version of the large face-width result for random 3-connected maps. These results are then used to derive…

We prove that if $\Gamma$ is a lattice in the group of isometries of a symmetric space of non-compact type without euclidean factors, then the virtual cohomological dimension of $\Gamma$ equals its proper geometric dimension.

Let $G$ be a connected complex semisimple Lie group, $\Gamma$ be a cocompact, irreducible and torsionless lattice in $G$ and $K$ be a maximal compact subgroup of $G$. Assume $\Gamma$ acts by left multiplication and $K$ acts by right…

Let L be a lattice in a connected Lie group. We show that besides a few exceptional cases, the deficiency of L is nonpositive.

We show that for any finitely presented group $G$, there is a simply connected closed 4-manifold containing an infinite family of topologically isotopic but smoothly inequivalent 2-links whose 2-link group is $G$. We also show that, if $G$…

This self-contained paper is part of a series \cite{FF2,FF3} on actions by diffeomorphisms of infinite groups on compact manifolds. The two main results presented here are: 1) Any homomorphism of (almost any) mapping class group or…

Let $M$ be complex projective manifold, and $A$ a positive line bundle on it. Assume that a compact and connected Lie group $G$ acts on $M$ in a Hamiltonian manner, and that this action linearizes to $A$. Then there is an associated unitary…

We show that for a typical high rank arithmetic lattice $\Gamma$, there exist finite index subgroups $\Gamma_{1}$ and $\Gamma_{2}$ such that $\Gamma_{1} \not\simeq \Gamma_{2}$ while $\widehat{\Gamma_{1}} \simeq \widehat{\Gamma_{2}}$. But…

A remarkable result of Thompson states that a finite group is soluble if and only if its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory…

A flat complete causal Lorentzian manifold is called {\it strictly causal} if the past and the future of each its point are closed near this point. We consider strictly causal manifolds with unipotent holonomy groups and assign to a…

We prove that, if $\Gamma$ is a finite connected cubic vertex-transitive graph, then either there exists a semiregular automorphism of $\Gamma$ of order at least $6$, or the number of vertices of $\Gamma$ is bounded above by an absolute…