Related papers: The generalised word problem for subgroups of hype…

We construct torsion-free hyperbolic groups without unique product whose subgroups up to some given finite index are themselves non-unique product groups. This is achieved by generalising a construction of Comerford to graphical small…

We define a class $\mathcal{U}$ of solvable groups of finite abelian section rank which includes all such groups that are virtually torsion-free as well as those that are finitely generated. Assume that $G$ is a group in $\mathcal{U}$ and…

A finitely generated group or monoid is said to be context-free if it has context-free word problem. In this note, we give an example of a context-free monoid, none of whose maximal subgroups are finitely generated. This answers a question…

A finitely generated group is lacunary hyperbolic if one of its asymptotic cones is an $\mathbb{R}$-tree. In this article we give a necessary and sufficient condition on lacunary hyperbolic groups in order to be stable under free product by…

We prove that the rank problem is decidable in the class of torsion-free word-hyperbolic Kleinian groups. We also show that every group in this class has only finitely many Nielsen equivalence classes of generating sets of a given…

We show that one can define and effectively compute Stallings graphs for quasi-convex subgroups of automatic groups (\textit{e.g.} hyperbolic groups or right-angled Artin groups). These Stallings graphs are finite labeled graphs, which are…

Let S be a closed surface of genus at least 2. We show that a finitely generated group G which is an extension of the fundamental group H of S is word hyperbolic if and only the orbit map of the quotient group G/H on the complex of curves…

We present some results about quasiconvex subgroups of infinite index and their products. After that we extend the standard notion of a subgroup commensurator to an arbitrary subset of a group, and generalize some of the previously known…

We give a new, effective proof of the separability of cubically convex-cocompact subgroups of special groups. As a consequence, we show that if $G$ is a virtually compact special hyperbolic group, and $Q\leq G$ is a $K$-quasiconvex…

We develop a theory of convex cocompact subgroups of the mapping class group MCG of a closed, oriented surface S of genus at least 2, in terms of the action on Teichmuller space. Given a subgroup G of MCG defining an extension L_G: 1-->…

For a finitely generated group, there are two recent generalizations of the notion of a quasiconvex subgroup of a word-hyperbolic group, namely a stable subgroup and a Morse or strongly quasiconvex subgroup. Durham and Taylor defined…

We study a characteristic subgroup of finitely generated groups, consisting of elements with uniform upper bound for word-lengths. For a group $G$, we denote this subgroup by $G_{bound}$. We give sufficient criteria for triviality and…

We show that the theory of the free group -- and more generally the theory of any torsion-free hyperbolic group -- is $n$-ample for any $n\geq 1$. We give also an explicit description of the imaginary algebraic closure in free groups.

There are many open questions surrounding the characterisation of groups with context-sensitive word problem. Only in 2018 was it shown that all finitely generated virtually Abelian groups have multiple context-free word problems, and it is…

We prove that it is decidable whether or not a finitely generated submonoid of a virtually free group is graded, introduce a new geometric characterization as quasi-geodesic monoids, and show that their word problem is rational (as a…

We study the language-theoretic aspects of the word problem, in the sense of Duncan & Gilman, of free products of semigroups and monoids. First, we provide algebraic tools for studying classes of languages known as super-AFLs, which…

In this paper we study hyperbolicty of the universal group $U(P)$ of a pregroup $P$. Given a finitely generated group $G$ and a pregroup $P$ such that $G \simeq U(P)$, we provide a particular set of axioms on $P$ which ensure that $G$ is…

Let $\mathcal G$ denote the space of finitely generated marked groups. We give equivalent characterizations of closed subspaces $\mathcal S\subseteq \mathcal G$ satisfying the following zero-one law: for any sentence $\sigma$ in the…

We describe solutions of the equation $x^ny^m=a^nb^m$ in acylindrically hyperbolic groups (AH-groups), where $a,b$ are non-commensurable special loxodromic elements and $n,m$ are integers with sufficiently large common divisor. Using this…

We prove an equivariant version of the fact that word-hyperbolic groups have finite asymptotic dimension. This is important in connection with our forthcoming proof of the Farrell-Jones conjecture in algebraic K-theory for every…