Related papers: On the cuspidalization problem for hyperbolic curv…

Given two hyperbolic curves over p-adic local fields, the absolute anabelian conjecture claims that any isomorphism between their \'etale fundamental group comes from an isomorphism of schemes. This conjecture was proven by S. Mochizuki for…

We construct compact hyperbolic 3-manifolds with totally geodesic boundary, such that the closed 3-pseudomanifolds obtained by coning off the boundary components are negatively curved and contain locally convex subspaces whose fundamental…

Let $$1 \to H \to G \to Q \to 1$$ be an exact sequence where $H= \pi_1(S)$ is the fundamental group of a closed surface $S$ of genus greater than one, $G$ is hyperbolic and $Q$ is finitely generated free. The aim of this paper is to provide…

We study the structure of the \'etale fundamental groups of smooth curves over certain arithmetic schemes, and investigate the relative version of Grothendieck's anabelian conjecture in this setting. Consequently, every hyperbolic curve…

We give a sufficient condition under which the fundamental group of a reglued graph of surfaces is hyperbolic. A reglued graph of surfaces is constructed by cutting a fixed graph of surfaces along the edge surfaces, then regluing by…

Part 1 : We remark that the conjugacy problem for pairs of hyperbolic au- tomorphisms of a finitely presented group (typically a free group) is decidable. The solution that we propose uses the isomorphism problem for the suspensions, and…

This paper aims to establish the geometrical finiteness for the natural isometric actions of (birational) automorphism groups on the hyperbolic spaces for K3 surfaces, Enriques surfaces, Coble surfaces, and irreducible symplectic varieties.…

A hyperbolic algebraic curve is a bounded subset of an algebraic set. We study the function theory and functional analytic aspects of these sets. We show that their function theory can be described by finite codimensional subalgebras of the…

We discuss how non-commutative fundamental groups could eventually contribute to algorithms for finding rational points on hyperbolic curves.

In this paper we prove that given a non-isotrivial family of hyperbolic curves in positive characteristic, the isomorphism type of the geometric fundamental group is not constant on the fibres of the family.

We obtain explicit formulas for the number of non-isomorphic elliptic curves with a given group structure (considered as an abstract abelian group). Moreover, we give explicit formulas for the number of distinct group structures of all…

We suggest a new approach to the study of relatively hyperbolic groups based on relative isoperimetric inequalities. Various geometric, algebraic, and algorithmic properties are discussed.

We study unions of fundamental domains of a Fuchsian group, especially those with hyperbolic plane metric realizing the metric of the corresponding hyperbolic surface. We call these unions the \textit{geodesic covers} of the Fuchsian group…

Given a finite graph of relatively hyperbolic groups with its fundamental group relatively hyperbolic and edge groups quasi-isometrically embedded and relatively quasiconvex in vertex groups, we prove that vertex groups are relatively…

This arXived paper has two independant parts, that are improved and corrected versions of different parts of a single paper once named "On equations in relatively hyperbolic groups". The first part is entitled "Existential questions in…

We prove that, given a torsion-free relatively hyperbolic group G with non-relatively-hyperbolic peripherals, isomorphic finite index subgroups of G have the same index. This applies for instance to fundamental groups of finite-volume…

Many geometric structures associated to surface groups can be encoded in terms of invariant cross ratios on their circle at infinity; examples include points of Teichm\"uller space, Hitchin representations and geodesic currents. We add to…

Gromov Hyperbolic groups have remarkable finiteness properties;for example those that are torsion-free are fundamental groups of finitecomplexes whose universal cover iscontractible (property~$F$). In this talk we will show thattheir…

This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is…

We establish an analog of a theorem of Stallings which asserts the homomorphisms between the universal nilpotent quotients induced by a homomorphism $G \to H$ of groups are isomorphisms provided a pair of homological conditions are…