arXiv research

New lattices in higher dimensions have dense surface subgroups.

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

Generic Hitchin representations generate dense subgroups.

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

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 …

The paper finds dense subgroups in certain Lie groups.

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

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

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

A dense amalgam connects boundaries of groups split by finite subgroups.

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

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

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…

Cohomology of Lie group quotient equals Lie algebra cohomology.

Proves unique maps from certain spaces to others.

New proof shows almost all surface group actions are dense.

We consider dense 2-generator multiplicative subgroups in $\mathbb C$ and show that for each point $z\in \mathbb C$ the set of limit values for the arguments of the powers of each generator at the point $z$ is either finite or is $[-π,π]$

New subgroup found in Lie groups with unusual properties.

Research examines coamenable subgroups in higher rank groups.

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

The Greenberg-Shalom hypothesis connects subgroup properties to lattice structures in Lie groups.

We show that the topological groups $Diff_{+}^{1}(I)$ and $Diff_{+}^{1}(\mathbb{S}^1)$ of orientation-preserving $C^1$-diffeomorphisms of the interval and the circle, respectively, admit finitely generated dense subgroups. We also investigate the question of genericity (in the sense of Baire category) of such finite to…

Study answers arithmeticity question for normal subgroup of lattices.

New space of currents defined for nonabelian free groups and malnormal subgroups.

Study thin hyperbolic reflection groups and their properties.

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

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 on horospheres in higher rank homogeneous spaces, proving density properties.

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

We construct CAT(0) groups containing subgroups whose Dehn functions are given by $x^s$, for a dense set of numbers $s \in [2, \infty)$. This significantly expands the known geometric behavior of subgroups of CAT(0) groups.

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

Paper finds surface groups can deform in reductive symmetric spaces.

Study subgroups preserving proper domains in flag manifolds.

New CAT(0) groups show superexponential subgroup Dehn functions.

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…

Maximal representations in symplectic lattices proven for most cases.

The paper shows BNSR-invariants of McCool groups are either dense or empty.

Let L be a Lie group and Lambda a lattice in L. Suppose G is a non-compact simple Lie group realized as a Lie subgroup of L, and the image of G on L/Lambda is dense. Let c be a diagonalizable element of G not contained in a compact subgroup. Let U be the expanding horospherical subgroup of G associated to c. Let Omega …

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.

Let A denote the algebraic closure of the rationals Q in the complex numbers C. Suppose G is a torsion-free group which contains a congruence subgroup as a normal subgroup of finite index and denote by U(G) the C-algebra of closed densely defined unbounded operators affiliated to the group von Neumann algebra. We prove…

Paper proves non-arithmetic Teichmüller length spectra for subgroup of mapping class groups.

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…

Study shows mixing of flows on specific geometric spaces.

The study examines growth of quadratic forms under Anosov subgroups.

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

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

We study cocompact lattices with dense projections in a product $G_1 \times G_2$ of locally compact groups and show, under the assumption that each $G_i$ is a closed subgroup of the automorphism group $Aut(T_i)$ of a regular tree satisfying certain local transitivity conditions, that such a lattice is contained in only…