Related papers: A note on the Farrell-Jones conjecture for relativ…

We prove the Farrell-Jones fibered isomorphism conjecture for several classes of Artin groups of finite and affine types. As a consequence, we compute explicitly the surgery obstruction groups of the finite type pure Artin groups.

We show that a one-ended simply connected at infinity hyperbolic group $G$ with enough codimension-1 surface subgroups has $\partial G \cong \mathbb{S}^2$. Combined with a result of Markovic, our result gives a new characterization of…

We construct an example of a torsion free freely indecomposable finitely presented non-quasiconvex subgroup $H$ of a word hyperbolic group $G$ such that the limit set of $H$ is not the limit set of a quasiconvex subgroup of $G$. In…

The goal of this paper is to give a conjectural census of complex hyperbolic sporadic groups. We prove that only finitely many of these sporadic groups are lattices. We also give a conjectural list of all lattices among sporadic groups, and…

We give a new proof of Quillen's conjecture for solvable groups via a geometric and explicit method. For p-solvable groups, we provide both a new proof using the Classification of Finite Simple Groups and an asymptotic version without…

We prove that a finitely generated group $G$ hyperbolic relative to the collection of finitely generated subgroups H_1,..., H_m has the Rapid Decay property if and only if each H_i, i=1,2,..., m, has the Rapid Decay property.

We prove the Grothendieck-Serre conjecture for quasi-split reductive groups schemes. Our method involves reducing to the Borel subgroup in order to conclude the result from purity for tori and the structure theorem for unipotent radicals of…

Let $G$ be a group and $H$ a subgroup of $G$. This note introduces an equivalent definition of hyperbolic embedded subgroup based on Bowditch's approach to relatively hyperbolic groups in terms of fine graphs.

We state and prove a combination theorem for relatively hyperbolic groups seen as geometrically finite convergence groups. For that, we explain how to contruct a boundary for a group that is an acylindrical amalgamation of relatively…

We call a finite group G ultrasolvable if it has a characteristic subgroup series whose factors are cyclic. It was shown by Durbin--McDonald that the automorphism group of an ultrasolvable group is supersolvable. The converse statement was…

We study the set of homomorphisms from a fixed finitely generated group into a family of groups which are `uniformly acylindrically hyperbolic'. Our main results reduce this study to sets of homomorphisms which do not diverge in an…

Let N be a finitely generated normal subgroup of a Gromov hyperbolic group G. We establish criteria for N to have solvable conjugacy problem and be conjugacy separable in terms of the corresponding properties of G/N. We show that the…

We provide a general sufficient condition for extendability of quasimorphisms on subgroups. This condition recovers the result of Hull--Osin on quasimorphisms on hyperbolically embedded subgroups, and the proof given in this paper is much…

We prove that for a free product $G$ with free factor system $\mathcal{G}$, any automorphism $\phi$ preserving $\mathcal{G}$, atoroidal (in a sense relative to $\mathcal{G}$) and none of whose power send two different conjugates of…

We classify finitely generated, residually finite automorphism-induced HNN-extensions in terms of the residual separability of a single associated subgroup. This classification provides a method to construct automorphism-induced…

The asymptotic bound for a length-based attack on the Conjugacy Search Problem in relatively hyperbolic groups is cubic for hyperbolic elements and a "small" polynomial for parabolic elements, depending on the Conjugacy Search Problem for…

Suppose that $G$ is a simple adjoint reductive group over $\mathbf{Q}$, with an exceptional Dynkin type, and with $G(\mathbf{R})$ quaternionic (in the sense of Gross-Wallach). Then there is a notion of modular forms for $G$, anchored on the…

We show that the subgroup lattice of any finite group satisfies Frankl's Union-Closed Conjecture. We show the same for all lattices with a modular coatom, a family which includes all supersolvable and dually semimodular lattices. A common…

Let $G$ be a finite group and $\alpha(G)=\frac{|C(G)|}{|G|}$\,, where $C(G)$ denotes the set of cyclic subgroups of $G$. In this short note, we prove that $\alpha(G)\leq\alpha(Z(G))$ and we describe the groups $G$ for which the equality…

We introduce the notion of finite stature of a family $\{H_i\}$ of subgroups of a group $G$. We investigate the separability of subgroups of a group $G$ that splits as a graph of hyperbolic special groups with quasiconvex edge groups. We…