Related papers: Thin hyperbolic reflection groups

For a free group $\mathbb{F}$ of finite rank such that $\text{rank}(\mathbb{F})\geq 3$, we prove that the set of weak limits of a conjugacy class in $\mathbb{F}$ under iterates of some hyperbolic $\phi\in\mathsf{Out(\mathbb{F})}$ is equal…

Let $\Lambda<SL(2,\mathbb{Z})$ be a finitely generated, non-elementary Fuchsian group of the second kind, and $v, w$ be two primitive vectors in $\mathbb{Z}^2-(0,0)$. We consider the set $\mathcal{S}=\{\langle…

Compact connected abelian groups, or protori, have intrinsic structural characteristics that present for the entire category. In the case of finite-dimensional torus-free protori, The Resolution Theorem for Compact Abelian Groups sets the…

We prove that in a cocompact complex hyperbolic arithmetic lattice $\Gamma < {\rm PU}(m,1)$ of the simplest type, deep enough finite index subgroups admit plenty of homomorphisms to $\mathbb{Z}$ with kernel of type $\mathscr{F}_{m-1}$ but…

This paper proves under certain conditions the existence of an algorithm, which detects relatively quasiconvex subgroups $H$ of relatively hyperbolic groups $(G,\mathbb{P})$. Additionally, this algorithm outputs an induced peripheral…

We obtain a number of finiteness results for groups acting on Gromov-hyperbolic spaces. In particular we show that a torsion-free locally quasiconvex hyperbolic group has only finitely many conjugacy classes of $n$-generated one-ended…

The article contains a survey of our results on weakly commensurable arithmetic and general Zariski-dense subgroups, length-commensurable and isospectral locally symmetric spaces and of related problems in the theory of semi-simple agebraic…

We give an algorithm that decides whether a single equation in a group that is virtually a class $2$ nilpotent group with a virtually cyclic commutator subgroup, such as the Heisenberg group, admits a solution. This generalises the work of…

We prove that finite lamplighter groups $\{\mathbb{Z}_2\wr\mathbb{Z}_n\}_{n\ge 2}$ with a standard set of generators embed with uniformly bounded distortions into any non-superreflexive Banach space, and therefore form a set of test-spaces…

Let $\Gamma$ be a finitely generated group which is hyperbolic relative to a finite family $\{H_1,...,H_n\}$ of subgroups. We prove that $\Gamma$ is uniformly embeddable in a Hilbert space if and only if each subgroup $H_i$ is uniformly…

Let $\Gamma$ be a nonelementary discrete subgroup of SU(n,1) or Sp(n,1). We show that if the trace field of $\Gamma$ is contained in $\mathbb R$, $\Gamma$ preserves a totally geodesic submanifold of constant negative sectional curvature.…

We introduce a new class of reflection groups associated with the canonical bilinear lattices of Lenzing, which we call reflection groups of canonical type. The main result of this work is a categorification of the corresponding poset of…

Discriminantal arrangements are hyperplane arrangements, which are generalized braid ones. They are constructed from given hyperplane arrangements, but their combinatorics are not invariant under combinatorial equivalence. However, it is…

The ordinary Bondi-Metzner-Sachs (BMS) group B is the common asymptotic symmetry group of all asymptotically flat Lorentzian space-times. As such, B is the best candidate for the universal symmetry group of General Relativity. However, in…

We provide the first examples of strongly dense representations of a hyperbolic 3-manifold group into $SL(4,\mathbb{R})$ and $SU(3,1)$ i.e. representations where every pair of non-commuting elements has Zariski dense image. Our examples are…

This paper is concerned with discrete, uniform subgroups (lattices) of oscillator groups, which are certain semidirect products of the Heisenberg group and the additive group of real numbers. The present paper rectifies the uncertainties in…

For any n>1 we determine the uniform and nonuniform lattices of the smallest covolume in the Lie group Sp(n,1). We explicitly describe them in terms of the ring of Hurwitz integers in the nonuniform case with n even, respectively, of the…

We present a general method to compute a presentation for any cusped arithmetic hyperbolic lattice $\Gamma$, applying a classical result of Macbeath to a suitable $\Gamma$-invariant horoball cover of the corresponding symmetric space. As…

We present new computational results for symplectic monodromy groups of hypergeometric differential equations. In particular, we compute the arithmetic closure of each group, sometimes justifying arithmeticity. The results are obtained by…

We give an algorithm for solving equations and inequations with rational constraints in virtually free groups. Our algorithm is based on Rips classification of measured band complexes. Using canonical representatives, we deduce an algorithm…