Related papers: The Surface Group Conjecture: Cyclically Pinched a…

Let $\mathcal{C}$ be a class of groups. A group $X$ is said to be residually a $\mathcal{C}$-group (conjugacy $\mathcal{C}$-separable) if, for any elements $x,y \in X$ that are not equal (not conjugate in $X$), there exists a homomorphism…

Given a group $G = H_1 \ast_A H_2$ which is the free product of two finitely generated groups $H_1$ and $H_2$ with amalgamation over a cyclic subgroup $A$ which is malnormal in $G$, we study relations between the structure of its subgroups…

We prove that one-relator groups with torsion are hereditarily conjugacy separable. Our argument is based on a combination of recent results of Dani Wise and the first author. As a corollary we obtain that any quasiconvex subgroup of a…

Hans Zassenhaus conjectured that every torsion unit of the integral group ring of a finite group $G$ is conjugate within the rational group algebra to an element of the form $\pm g$ with $g\in G$. This conjecture has been disproved recently…

We prove a freeness theorem for low-rank subgroups of one-relator groups. Let $F$ be a free group, and let $w\in F$ be a non-primitive element. The primitivity rank of $w$, $\pi(w)$, is the smallest rank of a subgroup of $F$ containing $w$…

Gordon and Wilton recently proved that the double D of a free group F amalgamated along a cyclic subgroup C of F contains a surface group if a generator w of C satisfies a certain 3-manifold theoretic condition, called virtually…

In this paper we investigate Gromov's question: whether every one-ended word hyperbolic group contains a surface subgroup. The case of double groups is considered by studying the associated one relator groups. We show that the majority…

We show that Baumslag-Solitar groups are virtually 2-avoidable, that is, they admit finite index subgroup whose first homology is devoid of $\mathbb{Z}_2$ summand. We also prove virtual 2-avoidability for some other classes of one-relator…

Motivated by the notion of boundary for hyperbolic and $CAT(0)$ groups, M. Bestvina in "Local Homology Properties of Boundaries of Groups" introduced the notion of a (weak) $\mathcal Z$-structure and (weak) $\mathcal Z$-boundary for a group…

A cocompact lattice in a semisimple Lie group $G$ is a discrete subgroup $\Gamma$ such that the quotient $G/\Gamma$ is compact. Does such a lattice always contain a surface group, i.e. a subgroup isomorphic to the fundamental group of a…

We show that all finitely generated free-by-cyclic groups are conjugacy separable: if a finitely generated group $G$ surjects onto $\mathbb{Z}$ with free kernel, then for every pair of non-conjugate elements $g,h\in G$, there exists a…

Consider a one-ended word-hyperbolic group. If it is the fundamental group of a graph of free groups with cyclic edge groups then either it is the fundamental group of a surface or it contains a finitely generated one-ended subgroup of…

We prove that any one-relator group $G$ is the fundamental group of a compact Sasakian manifold if and only if $G$ is either finite cyclic or isomorphic to the fundamental group of a compact Riemann surface of genus g > 0 with at most one…

We settle an old conjecture of Karrass and Solitar by proving that a finitely generated subgroup of a non-trivial free product $G = A\ast B$ has finite index if and only if it intersects non-trivially each non-trivial normal subgroup of…

Let $G$ be the semidirect product $\Gamma \rtimes F_2$ where $\Gamma$ is either the free group $F_n$, $n > 1$ or the fundamental group $S_g$ of a closed surface of genus $g > 1$. We prove that $G$ is incoherent, solving two problems posed…

We follow in this paper a recent line of work, consisting in characterizing the periodically rigid finitely generated groups, i.e., the groups for which every subshift of finite type which is weakly aperiodic is also strongly aperiodic. In…

Previously, the authors proved that the presentation complex of a one-relator group $G$ satisfies a geometric condition called negative immersions if every two-generator, one-relator subgroup of $G$ is free. Here, we prove that one-relator…

A classical discovery known as Fenchel's conjecture and proved in the 1950s, shows that every co-compact Fuchsian group $F$ has a normal subgroup of finite index isomorphic to the fundamental group of a compact unbordered orientable…

We use the theory of group actions on profinite trees to prove that the fundamental group of a finite, 1-acylindrical graph of free groups with finitely generated edge groups is conjugacy separable. This has several applications: we prove…

We say that a finite group $G$ satisfies the independence property if, for every pair of distinct elements $x$ and $y$ of $G$, either $\{x,y\}$ is contained in a minimal generating set for $G$ or one of $x$ and $y$ is a power of the other.…