Related papers: Limits of quasifuchsian groups with small bending

Hyperbolic buildings are central objects in both hyperbolic geometry and geometric group theory, exhibiting a wide range of intriguing characteristics, especially with respect to group actions. In this paper, we develop the theory of…

We study a family of Bowen-Series-like maps associated to any finitely generated Fuchsian group of the first kind with at least one cusp. These maps act on the boundary of the hyperbolic plane in a piecewise manner by generators of the…

Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in…

We show that every quasiconformal contact foliation supports an invariant metric and characterise such foliations by the dynamical property of $C^1$-equicontinuity. We prove that a generalisation of the Weinstein conjecture holds for…

We show that if $A$ is a closed subset of the Heisenberg group whose vertical projections are nowhere dense, then the complement of $A$ is quasiconvex. In particular, closed sets which are null sets for the cc-Hausdorff $3$-measure have…

In this article we show that for any given Riemann surface $\Sigma$ of genus $g$, we can bound (from above) the renormalized volume of a (hyperbolic) Schottky group with boundary at infinity conformal to $\Sigma$ in terms of the genus and…

We give upper bounds on the principal curvatures of a maximal surface of nonpositive curvature in three-dimensional Anti-de Sitter space, which only depend on the width of the convex hull of the surface. Moreover, given a quasisymmetric…

It is well known that every quasi-Fuchsian manifold admits at least one closed incompressible minimal surface, and at most finitely many of them. In this paper, for any prescribed integer $N>0$, we construct a quasi-Fuchsian manifold which…

We use minimal (or CMC) surfaces to describe 3-dimensional hyperbolic, anti-de Sitter, de Sitter or Minkowski manifolds. We consider whether these manifolds admit ``nice'' foliations and explicit metrics, and whether the space of these…

We show that averages on geometrically finite Fuchsian groups, when embedded via a representation into a space of matrices, have a homogeneous asymptotic limit under appropriate scaling. This generalizes some of the results of Maucourant to…

Classifying groups up to quasi-isometry is a fundamental problem in geometric group theory. In the context of hyperbolic and relatively hyperbolic groups, one of the key invariants in this classification is the boundary at infinity. F.…

We prove several functional and geometric inequalities only assuming the linearity and a quantitative $\mathrm{L}^\infty$-to-Lipschitz smoothing of the heat semigroup in metric-measure spaces. Our results comprise a Buser inequality, a…

We investigate Lindel\"of and Koebe type boundary behavior results for bounded quasiregular mappings in $n$-dimensional Euclidean space. Our results give sufficient conditions for the existence of non-tangential limits at a boundary point.

We show that M\"uller's $L\log L$ bound $$F(Du)\geq 0,\,Du\in L^p_{\mathrm{loc}}(\mathbb{R}^n)\implies F(Du)\in L\log L_{\mathrm{loc}}(\mathbb{R}^n)$$ for $F =\det$ and $p=n$ holds for quasiconcave $F$ which are homogeneous of degree $p>1$.…

We consider an Orlicz space based cohomology for metric (measured) spaces with bounded geometry. We prove the quasi-isometry invariance for a general Young function. In the hyperbolic case, we prove that the degree one cohomology can be…

We characterize sequences of Kleinian surface groups with convergent subsequences in terms of the asymptotic behavior of the ending invariants of the associated hyperbolic 3-manifolds. Asymptotic behavior of end invariants in a convergent…

The aim of this short note is to provide a proof of the decidability of the generalized membership problem for relatively quasi-convex subgroups of finitely presented relatively hyperbolic groups, under some reasonably mild conditions on…

We show that simply connected toric hyperK\"ahler metrics of finite topological type with maximal volume growth are generically quasi-asymptotically conical, which allows us to compute explicitly their reduced $L^2$-cohomology groups. In…

We construct the first examples of finitely presented groups with quadratic Dehn function containing a finitely generated infinite torsion subgroup. These examples are "optimal" in the sense that the Dehn function of any such finitely…

In this paper, we study the conformal measures of a normal subgroup of a cocompact Fuchsian group. In particular, we relate the extremal conformal measures to the eigenmeasures of a suitable Ruelle operator. Using Ancona's theorem, adapted…