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A system of commutative hyperbolic complex numbers in 2 dimensions is studied in this paper. Exponential and trigonometric forms are obtained for these hyperbolic twocomplex numbers. Expressions are given for the elementary functions of…
In this article, we prove a combination theorem for a complex of relatively hyperbolic groups. It is a generalization of Martin's \cite{martin} work for combination of hyperbolic groups over a finite $M_K$-simplicial complex, where $k\leq…
We construct affine uniformly Lipschitz actions on $\ell^1$ and $L^1$ for certain groups with hyperbolic features. For acylindrically hyperbolic groups, our actions have unbounded orbits, while for residually finite hyperbolic groups and…
We give a new technique for constructing presentations by generators and relations for representations of groups like $SL_n(\mathbb{Z})$ and $Sp_{2g}(\mathbb{Z})$. Our results play an important role in recent work of the authors calculating…
Based on the concept of dual cones introduced by J.Opgenorth we give an algorithm to compute a generating system of the group of automorphisms of a lattice endowed with a hyperbolic bilinear form.
In 1996, Gersten proved that finitely presented subgroups of a word hyperbolic group of integral cohomological dimension 2 are hyperbolic. We use isoperimetric inequalities over arbitrary rings to extend this result to any ring. In…
We construct two-dimensional families of complex hyperbolic structures on disc orbibundles over the sphere with three cone points. This contrasts with the previously known examples of the same type, which are locally rigid. In particular,…
We look at group actions on metric spaces, particularly at group actions on geodesic hyperbolic spaces. We classify the types of automorphisms on these spaces and prove several results about the density of the hyperbolic limit set of the…
These notes are the English version of the paper "Hyperbolicit\'e du graphe des rayons et quasi-morphismes sur un gros groupe modulaire". The mapping class group Gamma of the complement of a Cantor set in the plane arises naturally in…
We show that a relatively hyperbolic group either is virtually cyclic or has uniform exponential growth.
Minor changes in the exposition and small corrections on the previous version.
In this paper, we consider the geodesic flow on factors of the hyperbolic plane. We prove that a periodic orbit including a 2-antiparallel encounter has a partner orbit. We construct the partner orbit and give an estimate for the action…
We generalize a result of Paulin on the Gromov boundary of hyperbolic groups to the Morse boundary of proper, maximal hierarchically hyperbolic spaces admitting cocompact group actions by isometries. Namely we show that if the Morse…
We show that the moduli space M of marked cubic surfaces is biholomorphic to the quotient by a discrete group generated by complex reflections of the complex four-ball minus the reflection hyperplanes of the group. Thus M carries a complex…
The extended mapping class group of a surface $\Sigma$ is defined to be the group of isotopy classes of (not necessarily orientation-preserving) homeomorphisms of $\Sigma$. We are able to show that the extended mapping class group of an…
The group of automorphisms of the free group on two generators is known to act geometrically, in an essentially unique way, on a 2-dimensional CAT(0) space X. We prove that X contains precisely two Hamiltonian surfaces. By this we mean a…
This paper finds the generators of the automorphism group of the group of unitriangular matrices over a field. Most of this paper is an exposition of the work of V.M. Lev\u{c}huk, part of which is in Russian. Some proofs are of my own.
We introduce a number of new tools for the study of relatively hyperbolic groups. First, given a relatively hyperbolic group G, we construct a nice combinatorial Gromov hyperbolic model space acted on properly by G, which reflects the…
We construct triangular hyperbolic polyhedra whose links are generalized 4-gons. The universal cover of those polyhedra are hyperbolic buildings, which appartments are hyperbolic planes tesselated by regular triangles with angles $\pi/4$.…
We prove that the topological complexity of a finite index subgroup of a hyperbolic group is linear in its index. This follows from a more general result relating the size of the quotient of a free cocompact action of hyperbolic group on a…