Related papers: All hyperbolic cyclically presented groups with po…

We show that the mapping torus of a hyperbolic group by a hyperbolic automorphism is cubulable. Along the way, we (i) give an alternate proof of Hagen and Wise's theorem that hyperbolic free-by-cyclic groups are cubulable, and (ii) extend…

We give a new and completely algebraic proof of the Helton-Vinnikov Theorem stating that every hyperbolic polynomial in three variables admits a definite linear determinantal representation.

In the framework of homological characterizations of relative hyperbolicity, Groves and Manning posed the question of whether a simply connected $2$-complex $X$ with a linear homological isoperimetric inequality, a bound on the length of…

Tits has defined Kac-Moody and Steinberg groups over commutative rings, providing infinite dimensional analogues of the Chevalley-Demazure group schemes. Here we establish simple explicit presentations for all Steinberg and Kac-Moody groups…

The purpose of this note is to provide a short alternate proof that (combined with a theorem proven by Szczepanski) shows that a group which is relatively hyperbolic in the sense of the definition of Gromov is relatively hyperbolic in the…

A projective manifold $M$ is algebraically hyperbolic if there exists a positive constant $A$ such that the degree of any curve of genus $g$ on $M$ is bounded from above by $A(g-1)$. A classical result is that Kobayashi hyperbolicity…

Let $G$ be an acylindrically hyperbolic group and $E$ an exponential equation over $G$. We show that if $E$ is solvable in $G$, then there exists a solution whose components, corresponding to loxodromic elements, can be linearly estimated…

Our first result gives a partial converse to a well-known theorem of A. Ancona for hyperbolic groups. We prove that a group $G$, equipped with a symmetric probability measure whose finite support generates $G$, is hyperbolic if it is…

In this article, I study some classes of finitely presented groups with the aim of finding out whether the maximal metabelian quotients of the members of these classes admit finite presentations. The considered classes include those of…

We give a dynamical characterization of acylindrically hyperbolic groups. As an application, we prove that non-elementary convergence groups are acylindrically hyperbolic.

In this note, we prove that a random extension of either the free group $F_N$ of rank $N\ge3$ or of the fundamental group of a closed, orientable surface $S_g$ of genus $g\ge2$ is a hyperbolic group. Here, a random extension is one…

A hyperbolic reflection group is a discrete group generated by reflections in the faces of an $n$-dimensional hyperbolic polyhedron. This survey article is dedicated to the study of arithmetic hyperbolic reflection groups with an emphasis…

We consider the class $\mathcal{R}$ of finitely generated toral relatively hyperbolic groups. We show that groups from $\mathcal{R}$ are commutative transitive and generalize a theorem proved by Benjamin Baumslag to this class. We also…

Following the previous work of Nikulin and Agol, Belolipetsky, Storm, and Whyte it is known that there exist only finitely many (totally real) number fields that can serve as fields of definition of arithmetic hyperbolic reflection groups.…

In this paper we study the discreteness of complex hyperbolic triangle groups of type $[m,m,0; n_1, n_2, 2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $n_1, n_2, 2$ in…

In this paper, we mainly study hyperbolic semigroups from which we get non-empty escaping set and Eremenko's conjecture remains valid. We prove that if each generator of bounded type transcendental semigroup S is hyperbolic, then the…

We construct a sequence of balanced finite presentations of the trivial group with two generators and two relators with the following property: The minimal number of relations required to demonstrate that a generator represents the trivial…

We show that one relator groups viewed as metric spaces with respect to the word-length metric have finite asymptotic dimension in the sense of Gromov and give an estimate of their asymptotic dimension in terms of the relator length.

A subset of a group is characteristic if it is invariant under every automorphism of the group. We study word length in fundamental groups of closed hyperbolic surfaces with respect to characteristic generating sets consisting of a finite…

We give formulas for the numbers of type II and type IV hyperbolic components in the space of quadratic rational maps, for all fixed periods of attractive cycles.