arXiv research

New lattices in higher dimensions have dense surface subgroups.

Bi-Lipschitz rigidity theorem for dense subgroups of algebraic groups.

The paper finds free semigroups in dense subgroups of Lie groups with critical exponents arbitrarily close to the subgroup's.

The paper finds dense subgroups in certain Lie groups.

The study finds conditions for certain groups to be dense in a specific mathematical space.

We begin by showing that commensurators of Zariski dense subgroups of isometry groups of symmetric spaces of non-compact type are discrete provided that the limit set on the Furstenberg boundary is not invariant under the action of a (virtual) simple factor. In particular for rank one or simple Lie groups, Zariski dens…

Generic Hitchin representations generate dense subgroups.

Regular subgroups of SL3(R) are identified and ruled out.

Proves unique maps from certain spaces to others.

Odd-dimensional SL(n,Q) contains dense surface subgroups.

New subgroup found in Lie groups with unusual properties.

The study explores deformations of discrete subgroups in non-compact homogeneous spaces.

Research examines coamenable subgroups in higher rank groups.

Maximal representations in symplectic lattices proven for most cases.

The paper proves a unique conformal measure for Anosov groups and shows local mixing.

Let ${{\bf H}_{\mathbb H}}^n$ denote the $n$-dimensional quaternionic hyperbolic space. The linear group ${\rm{Sp}}(n,1)$ acts by the isometries of ${{\bf H}_{\mathbb H}}^n$. A subgroup $G$ of ${\rm {Sp}}(n,1)$ is called \emph{Zariski dense} if it does not fix a point on ${{\bf H}_{\mathbb H}}^n \cup \partial {{\bf H}_…

Study thin hyperbolic reflection groups and their properties.

Study critical exponents in normal subgroups of higher rank Lie groups.

Paper finds surface groups can deform in reductive symmetric spaces.

Example shows dense subgroup of SL5(Z) not finitely presented.

Let $\mathbb F=\mathbb R$, $\mathbb C$ or $\mathbb H$. Let ${\bf H}_{\mathbb F}^n$ denote the $n$-dimensional $\mathbb F$-hyperbolic space. Let ${\rm U}(n,1; \mathbb F)$ be the linear group that acts by the isometries. A subgroup $G$ of ${\rm U}(n,1; \mathbb F)$ is called \emph{Zariski dense} if it does not fix a point…

Study subgroups preserving proper domains in flag manifolds.

Extends Hopf-Tsuji-Sullivan dichotomy to higher rank groups and applies to Anosov subgroups.

Let $G$ be a simply connected, solvable Lie group and $Γ$ a lattice in $G$. The deformation space $\mathcal{D}(Γ,G)$ is the orbit space associated to the action of $\Aut(G)$ on the space $\mathcal{X}(Γ,G)$ of all lattice embeddings of $Γ$ into $G$. Our main result generalises the classical rigidity theorems of Mal'tsev…

The article contains a survey of results on length-commensurable and isospectral locally symmetric spaces and related problems in the theory of semi-simple algebraic groups.

New representations of hyperbolic 3-manifold groups into larger groups.

The study proposes a conjecture about the monodromy group of singular hyperbolic metrics and provides evidence and confirmations.

We show that every limit point of a Zariski dense discrete subgroup $Γ$ of the isometry group of a symmetric space of noncompact type is conical if and only if $Γ$ is convex cocompact.

Study shows mixing of flows on specific geometric spaces.

The study examines growth of quadratic forms under Anosov subgroups.

New domains of discontinuity found for Anosov representations.

The purpose of this article is to present a survey of our recent results on length commensurable and isospectral locally symmetric spaces. The geometric questions led us to the notion of "weak commensurability" of two Zariski-dense subgroups in a semi-simple Lie group. We have shown that for arithmetic subgroups, weak …

The paper bounds growth indicator functions for discrete subgroups in algebraic groups.

Abelian covers of hyperbolic $3$-manifolds are ubiquitous. We prove the local mixing theorem of the frame flow for abelian covers of closed hyperbolic $3$-manifolds. We obtain a classification theorem for measures invariant under the horospherical subgroup. We also describe applications to the prime geodesic theorem as…

Let G=SO(n,1) and Gamma a geometrically finite Zariski dense subgroup of G which is contained in an arithmetic subgroup of G. Denoting by Gamma(q) the principal congruence subgroup of Gamma of level q, and fixing a positive number λ_0 strictly smaller than (n-1)^2/4, we show that, as q tends to infinity along primes, t…

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

In various classes of infinite groups, we identify groups that are presentable by products, i.e. groups having finite index subgroups which are quotients of products of two commuting infinite subgroups. The classes we discuss here include groups of small virtual cohomological dimension and irreducible Zariski dense sub…

The study finds discrete subgroups with full limit sets in higher rank Lie groups.

The paper counts conjugacy classes of loxodromic elements in Anosov subgroups with a power saving error term.

The study proves conditions for rigidity of Kleinian groups using measure theory and ergodic theory.

We build examples of properly convex projective manifold $Ω/ Γ$ which have finite volume, are not compact, nor hyperbolic in every dimension $n \geqslant 2$. On the way, we build Zariski-dense discrete subgroups of $\SL_{n+1}(\R)$ which are not lattice, nor Schottky groups. Moreover, the open properly convex set $Ω$ is…

The paper studies invariant measures for specific actions in algebraic groups.

Let SL(2, $\mathbb H$) be the group of $2 \times 2$ quaternionic matrices $A=\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ with quaternionic determinant $\det A=|ad-aca^{-1} b|=1$. This group acts by the orientation-preserving isometries of the five dimensional (real) hyperbolic space. We obtain discreteness criteria f…

Anosov subgroup equidistributes geodesics and holonomies on homogeneous spaces.

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 groups. We have included a discussion of very recent results and conjectures on…

In this note, we study deformations of discrete and Zariski dense subgroups of SU(2, 1) in quaternionic hyperbolic space. Specifi- cally we consider two examples coming from representations of 3-manifold groups (the figure eight knot and Whitehead links complement) and show opposite behavior: one is not deformable outs…

Anosov groups' measures on limit sets are uniquely determined by their dimension.

Study measures invariant under horospherical subgroups for finitely generated Kleinian groups.