Related papers: Counting problems for special-orthogonal Anosov re…

For $\Gamma$ a cofinite Fuchsian group, and $l$ a fixed closed geodesic, we study the asymptotics of the number of those images of $l$ that have a prescribed orientation and distance from $l$ less than or equal to $X$. Using a new relative…

We build an analogue of the Gromov boundary for any proper geodesic metric space, hence for any finitely generated group. More precisely, for any proper geodesic metric space $X$ and any sublinear function $\kappa$, we construct a boundary…

Let $n\in\mathbb{N}$ and let $\Theta \subset \{1,\dots,n\}$ be a non-empty subset. We prove that if $\Theta$ contains an odd integer, then any $P_\Theta$-Anosov subgroup of ${\rm Sp}(2n,\mathbb{R})$ is virtually isomorphic to a free group…

We prove that a word hyperbolic group whose Gromov boundary properly contains a $2$-sphere cannot admit a projective Anosov representation into $\mathsf{Sp}_{2m}(\mathbb{C})$, $m\in \mathbb{N}$. We also prove that a word hyperbolic group…

We give the asymptotic growth of the number of (multi-)arcs of bounded length between boundary components on complete finite-area hyperbolic surfaces with boundary. Specifically, if $S$ has genus $g$, $n$ boundary components and $p$…

We show that every non-elementary group $G$ acting properly and cocompactly by isometries on a proper geodesic Gromov hyperbolic space $X$ is growth tight. In other words, the exponential growth rate of $G$ for the geometric…

We show that any infinite order element $g$ of a virtually cyclic hyperbolically embedded subgroup of a group $G$ is Morse, that is to say any quasi-geodesic connecting points in the cyclic group $C$ generated by $g$ stays close to $C$.…

Let G be a connected semisimple Lie group such that the associated symmetric space X is Hermitian and let Gamma be the fundamental group of a compact orientable surface of genus at least 2. We survey the study of maximal representations,…

Botelho, Jamison, and Moln\'ar \cite{BJM}, and Geh\' er and \v{S}emrl \cite{GeS} have recently described the general form of surjective isometries of Grassmann spaces of all projections of a fixed finite rank on a Hilbert space $H$. As a…

This paper treats mathematically some problems in p-adic quantum mechanics. We first deal with p-adic symplectic group corresponding to the symmetry on the classical phase space. By the filtrations of isotropic subspaces and almost…

We study the limit set of discrete subgroups arising from Anosov representations. Specially we study the limit set of discrete groups arising from strictly convex real projective structures and Anosov representations from a finitely…

In arXiv:1403.7671, Kapovich, Leeb and Porti gave several new characterizations of Anosov representations $\Gamma \to G$, including one where geodesics in the word hyperbolic group $\Gamma$ map to "Morse quasigeodesics" in the associated…

We propose a definition for the length of closed geodesics in a globally hyperbolic maximal compact (GHMC) Anti-De Sitter manifold. We then prove that the number of closed geodesics of length less than $R$ grows exponentially fast with $R$…

It is a longstanding problem to determine the precise relationship between the geodesic length spectrum of a hyperbolic manifold and its commensurability class. A well known result of Reid, for instance, shows that the geodesic length…

We show that the action on its orbit space induced by a pseudo-Anosov flow on a closed $3$-manifold (and more general Anosov-like actions) can be seen as an isometric action on a Gromov-hyperbolic space. When the flow is not $\R$-covered,…

We give an explicit solution of a q-Riemann Hilbert problem which arises in the theory of orthogonal polynomials, prove that it is unique, and deduce several properties. Our new results include the asymptotic behaviour of zeroes in the…

In this paper, we study orbit counting problems for inner functions using geodesic and horocyclic flows on Riemann surface laminations. For a one component inner function of finite Lyapunov exponent with $F(0) = 0$, other than $z \to z^d$,…

A long-standing open problem in systolic geometry asks whether a Riemannian metric on the real projective space whose volume equals that of the canonical metric, but is not isometric to it, must necessarily carry a periodic geodesic of…

We study the topology of admissible-loop spaces on a step-two Carnot group G. We use a Morse-Bott theory argument to study the structure and the number of geodesics on G connecting the origin with a 'vertical' point (geodesics are critical…

In this paper we prove the existence of isoperimetric regions of any volume in Riemannian manifolds with Ricci bounded below assuming Gromov--Hausdorff asymptoticity to the suitable simply connected model of constant sectional curvature.…