arXiv research

New rigidity theorem for product of lattices.

The study finds non-uniform lattices with thin Hitchin representations in specific Lie groups.

Study investigates lattices fibring over the circle, focusing on BNSR invariants.

We show that two uniform lattices of a regular right-angled Fuchsian building are commensurable, provided the chamber is a polygon with at least six edges. We show that in an arbitrary Gromov-hyperbolic regular right-angled building associated to a graph product of finite groups, a uniform lattice is commensurable with…

Study complex hyperbolic lattices and their relation to strict hyperbolization.

This paper investigates the geometry of compact contact manifolds that are uniformized by contact Lie groups, i.e., compact manifolds that are the quotient of some Lie group G with a left invariant contact structure and a uniform lattice subgroup. We re-examine Alexander's criteria for existence of lattices on solvable…

We prove, under the assumption of the virtual fibration conjecture for arithmetic hyperbolic 3-manifolds, that all arithmetic lattices in O(n,1), n> 4, and different from 7, are non-coherent. We also establish noncoherence of uniform arithmetic lattices of the simplest type in SU(n,1), n> 1, and of uniform lattices in …

The paper finds dense subgroups in certain Lie groups.

Thin groups found in specific lattices.

We study the covolumes of arithmetic lattices in $PSL_2(\mathbb R)^n$ for $n\geq 2$ and identify uniform and non-uniform irreducible lattices of minimal covolume. More precisely, let $μ$ be the Euler-Poincaré measure on $PSL_2(\mathbb R)^n$ and $χ=μ/2^n$. We show that the Hilbert modular group $PSL_2(\mathfrak o_{k_{49…

The paper uses Tannakian reconstruction to understand hyperbolic log-orbi curves.

Non-uniform lattices in PU(n,1) cannot geometrically act on CAT(0) cube complexes.

The paper proves Zimmer's conjecture for non-uniform lattices by controlling mass escape and Lyapunov exponents.

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 icosian ring in the uniform case for all n>1.

We prove that if $G$ is a non-uniform lattice in a rank-one semi-simple Lie group $\ne Isom(\H^2_\R)$ then $G$ is quasi-isometrically co-Hopf. This means that every quasi-isometric embedding $G\to G$ is coarsely onto and thus is a quasi-isometry.

The study finds that certain hyperbolic manifolds contain subgroups isomorphic to surface groups.

We study representations of lattices of PU(m,1) into PU(n,1). We show that if a representation is reductive and if m is at least 2, then there exists a finite energy harmonic equivariant map from complex hyperbolic m-space to complex hyperbolic n-space. This allows us to give a differential geometric proof of rigidity …

In this note, we study deformations of a non-uniform real hyperbolic lattice in quaternionic hyperbolic spaces. Specially we show that the representations of the fundamental group of the figure eight knot complement into PU(2,1) cannot be deformed in $PSp(2,1)$ out of PU(2,1) up to conjugacy.

The paper finds incommensurable lattices in complex models of Baumslag-Solitar groups.

Proof of boundedness of quasimorphisms for certain Lie groups.

The paper proves rigidity for complex Kleinian groups.

The study explores maps of 2- and 3-uniform tilings on the torus.

We introduce a new discrete system that arises from ellipsoidal billiards and is closely related to the double reflection nets. The system is defined on the lattice of a uniform honeycomb consisting of rectified hypercubes and cross polytopes. In the $2$-dimensional case, the lattice is regular and it incorporates dyna…

Proof shows volumes of certain geometric representations are always integers.

Let $G$ and $G'$ be simple Lie groups of equal real rank and real rank at least $2$. Let $Γ<G$ and $Λ< G'$ be non-uniform lattices. We prove a theorem that often implies that any quasi-isometric embedding of $Γ$ into $Λ$ is at bounded distance from a homomorphism. For example, any quasi-isometric embedding of $SL(n,\ma…

Let $Γ$ be a lattice in a connected semisimple Lie group $G$ with trivial center and no compact factors. We introduce a volume invariant for representations of $Γ$ into $G$, which generalizes the volume invariant for representations of uniform lattices introduced by Goldman. Then, we show that the maximality of this vo…

Study on Čech cohomology of Morse boundaries in hyperbolic manifolds.

The fundamental group of a Riemannian manifold with $δ$-pinched negative curvature, $δ>1/4$, cannot be the fundamental group of a quasicompact Kähler manifold. The proof also implies that a non-uniform lattice in $F_{4(-20)}$ cannot be the fundamental group of a quasicompact Kähler manifold. We also construct examples …

Let $(W,S)$ be a Coxeter system with Davis complex $Σ$. The polyhedral automorphism group $G$ of $Σ$ is a locally compact group under the compact-open topology. If $G$ is a discrete group (as characterised by Haglund--Paulin), then the set $\mathcal V_u(G)$ of uniform lattices in $G$ is discrete. Whether the converse i…

We prove general superrigidity results for actions of irreducible lattices on CAT(0) spaces; first, in terms of the ideal boundary, and then for the intrinsic geometry (including for infinite-dimensional spaces). In particular, one obtains a new and self-contained proof of Margulis' superrigidity theorem for uniform ir…

Let $ρ$ be a maximal representation of a uniform lattice $Γ\subset{\rm SU}(n,1)$, $n\geq 2$, in a classical Lie group of Hermitian type $H$. We prove that necessarily $H={\rm SU}(p,q)$ with $p\geq qn$ and there exists a holomorphic or antiholomorphic $ρ$-equivariant map from complex hyperbolic space to the symmetric sp…

We complete the classification of maximal representations of uniform complex hyperbolic lattices in Hermitian Lie groups by dealing with the exceptional groups ${\rm E}_6$ and ${\rm E}_7$. We prove that if $ρ$ is a maximal representation of a uniform complex hyperbolic lattice $Γ\subset{\rm SU}(1,n)$, $n>1$, in an exce…

The paper constructs Anosov representations for specific types of groups.

We investigate the rank gradient and growth of torsion in homology in residually finite groups. As a tool, we introduce a new complexity notion for generating sets, using measured groupoids and combinatorial cost. As an application we prove the vanishing of the above invariants for Farber sequences of subgroups of righ…

The paper defines Benoist-Hulin groups and explores their properties.

In this paper, we address the issue of quaternionic Toledo invariant to study the character variety of two dimensional complex hyperbolic uniform lattices into $SU(n,2)$. We construct four distinct representations to prove that the character variety contains at least four distinct components. We also address the existe…

The moduli space of lattices of $\mathbb{C}$ is a Riemann surface of finite hyperbolic area with the square lattice as an origin. We select a lattice from the induced uniform distribution and calculate the statistics of the Teichmüller distance to the origin. This in turn identifies distribution of the distance in Teic…

Let $Γ$ be a non-uniform lattice in $PU(p,1)$ without torsion and with $p\geq2$. We introduce the notion of volume for a representation $ρ:Γ\rightarrow PU(m,1)$ where $m \geq p$. We use this notion to generalize the Mostow--Prasad rigidity theorem. More precisely, we show that given a sequence of representations $ρ_n:…

We show that uniform lattices in some semi-simple groups (notably complex ones) admit Anosov surface subgroups. This result has a quantitative version: we introduce a notion, called $K$-Sullivan maps, which generalizes the notion of $K$-quasi-circles in hyperbolic geometry, and show in particular that Sullivan maps are…

New lattices in higher rank contain a fixed 3-manifold group with increasing systole.

Paper proves rigidity of Doyle spirals in hexagonal lattice circle packings.

We prove that, among metrics on a compact quotient of $\left(\mathbb{H}^2\right)^n$ (product of hyperbolic planes) of prescribed total volume, the product of hyperbolic metrics has minimal volume entropy.

Let I(p,v) be Bourdon's building, the unique simply-connected 2-complex such that all 2-cells are regular right-angled hyperbolic p-gons and the link at each vertex is the complete bipartite graph K(v,v). We investigate and mostly determine the set of triples (p,v,g) for which there exists a uniform lattice Γ in Aut(I(…

New method approximates hyperbolic lattices using cube complexes.

Method computes harmonic and conformal maps from point clouds.

We study quasi-isometric embeddings of symmetric spaces and non-uniform irreducible lattices in semisimple higher rank Lie groups. We show that any quasi-isometric embedding between symmetric spaces of the same rank can be decomposed into a product of quasi-isometric embeddings into irreducible symmetric spaces. We thu…

New MCMC method samples from lattice distributions efficiently.

As for the theory of maximal representations, we introduce the volume of a Zimmer's cocycle $Γ\times X \rightarrow \mbox{PO}^\circ(n, 1)$, where $Γ$ is a torsion-free (non-)uniform lattice in $\mbox{PO}^\circ(n, 1)$, with $n \geq 3$, and $X$ is a suitable standard Borel probability $Γ$-space. Our numerical invariant ex…