arXiv research

Example found of subgroup not a lattice in product of Lie groups

The study examines discrete subgroups of Lie groups and their residual finiteness.

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

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

The paper proves that certain spaces have injective balls of any radius.

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

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…

The study examines discrete subgroups of PSL2 over non-archimedean fields.

Improved homological dimension for certain subgroups in Lie groups.

Study answers arithmeticity question for normal subgroup of lattices.

In this paper, we generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete subgroups $Γ$ of isometries of negatively pinched Hadamard manifolds $X$. We then generalize a theorem of Bishop to prove that every discrete geome…

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

Let Gamma < PSL_2(C) be discrete, cofinite volume, and noncocompact. We prove that for all K > 1, there is a subgroup H < Gamma that is K-quasiconformally conjugate to a discrete cocompact subgroup of PSL_2(R). Along with previous work of Kahn and Markovic, this proves that every finite covolume Kleinian group has a ne…

Proves critical exponent for $Θ-$positive representations in discrete subgroups.

We are raising questions on discrete and dense subgroups of Diff(I). Most of the questions are around the problems discussed in [A1]-[A4].

New spaces found without certain actions, using special subgroups.

The paper proves rigidity and ergodicity of horospherical foliations.

Proves finite measure implies product structure for certain discrete subgroups.

Survey on Coxeter groups for Lie group examples.

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

Study continuous paths in discrete subgroups of hyperbolic space, proving combination and decomposition theorems.

Paper tackles decidability of subgroup discreteness problem.

Margulis wrote in the preface of his book Discrete subgroups of semisimple Lie groups that "A number of important topics have been omitted. The most significant of these is the theory of Kleinian groups and Thurston's theory of 3-dimensional manifolds: these two theories can be united under the common title of Theory o…

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the assumption of bounded torsion, to the case of negatively pinched Hadamard manifolds. Eve…

The study of topological groups with compact open subgroups and their geometric properties.

We study infinite covolume discrete subgroups of higher rank semisimple Lie groups, motivated by understanding basic properties of Anosov subgroups from various viewpoints (geometric, coarse geometric and dynamical). The class of Anosov subgroups constitutes a natural generalization of convex cocompact subgroups of ran…

New subgroup found in Lie groups with unusual properties.

Anosov subgroups' deformations affect limit cones and growth indicators continuously.

Study thin hyperbolic reflection groups and their properties.

Irreducible groups cannot be free if they ergodically act on a boundary.

Combination theorems for convex projective geometry subgroups.

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}_…

We give an affirmative answer to many cases of a question due to Shalom, which asks if the commensurator of a thin subgroup of a Lie group is discrete. In this paper, let $K<Γ<G$ be an infinite normal subgroup of an arithmetic lattice $Γ$ in a rank one simple Lie group $G$, such that the quotient $Q=Γ/K$ is infinite. W…

MOB-dS uses permutation to correct for dependency in discrete survival data.

This survey is based on a series of lectures that we gave at MSRI in Spring 2015 and on a series of papers, mostly written jointly with Joan Porti. Our goal here is to: 1. Describe a class of discrete subgroups $Γ<G$ of higher rank semisimple Lie groups, which exhibit some "rank 1 behavior". 2. Give different character…

We propose several common extensions of the classes of Anosov subgroups and geometrically finite Kleinian groups among discrete subgroups of semisimple Lie groups. We relativize various dynamical and coarse geometric characterizations of Anosov subgroups given in our earlier work, extending the class from intrinsically…

Study compact plane waves, showing they are essentially standard.

For relatively hyperbolic groups, we investigate conditions guaranteeing that the subgroup generated by two relatively quasiconvex subgroups $Q_1$ and $Q_2$ is relatively quasiconvex and isomorphic to $Q_1 \ast_{Q_1 \cap Q_2} Q_2$. The main theorem extends results for quasiconvex subgroups of word-hyperbolic groups, an…

It is shown that a closed solvable subgroup of a connected Lie group is compactly generated. In particular, every discrete solvable subgroup of a connected Lie group is finitely generated. Generalizations to locally compact groups are discussed as far as they carry.

Let $H<\mathrm{PSL}_2(\mathbb{Z})$ be a finite index normal subgroup which is contained in a principal congruence subgroup, and let $Φ(H)\neq H$ denote a term of the lower central series or the derived series of $H$. In this paper, we prove that the commensurator of $Φ(H)$ in $\mathrm{PSL}_2(\mathbb{R})$ is discrete. W…

We construct an infinite discrete subgroup of the isometry group of $\mathbb H^3$ with no finite quotients other than the trivial group.

The paper encourages Kleinian group thinking for higher rank Lie groups.

We consider sequences of finitely generated discrete subgroups Gamma_i=rho_i(Gamma) of a rank 1 Lie group G, where the representations rho_i are not necessarily faithful. We show that, for algebraically convergent sequences (Gamma_i), unless Gamma_i's are (eventually) elementary or contain normal finite subgroups of ar…

We develop the structure theory of full isometry groups of locally compact non-positively curved metric spaces. Amongst the discussed themes are de Rham decompositions, normal subgroup structure and characterising properties of symmetric spaces and Bruhat--Tits buildings. Applications to discrete groups and further dev…

We prove that a dense subgroup of $\mathrm{Homeo}_{+}(I)$ is not elementary amenable. We also show that the topological group $\mathrm{Homeo}_{+}(I)$ does not satisfy the Stability of the Generators Property, moreover, any finitely generated subgroup of $\mathrm{Homeo}_{+}(I)$ admits a faithful discrete representation …

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 critical exponents in normal subgroups of higher rank Lie groups.

We study the geometry and dynamics of discrete infinite covolume subgroups of higher rank semisimple Lie groups. We introduce and prove the equivalence of several conditions, capturing "rank one behavior'' of discrete subgroups of higher rank Lie groups. They are direct generalizations of rank one equivalents to convex…