Related papers: Soluble groups with no $\mathbb{Z} \wr \mathbb{Z}$…

In this paper we prove that a free nilpotent group of finite rank is transitive self-similar. In contrast, we prove that a free metabelian group of rank $r \geq 2$ is not transitive self-similar.

Let X be a smooth compactification of a connected linear algebraic group over a field k. The Chow group of degree nought zero-cycles on X is a torsion group. When k is a p-adic field, we show that the prime-to-p component of this group is…

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…

A group is said to be cube-free if its order is not divisible by the cube of any prime. Let $f_{cf,sol}(n)$ denote the isomorphism classes of solvable cube-free groups of order $n$. We find asymptotic bounds for $f_{cf,sol}(n)$ in this…

We give several examples of finite groups $G$ for which the rank of the tensor product $\mathbb{Z} \otimes_{\mathbb{Z}\mathrm{Aut}(G)}$ Wh$(G)$ is or is not zero. This is motivated by an earlier theorem of the first author, which implies as…

The simplicial wedge construction on simplicial complexes and simple polytopes has been used by a variety of authors to study toric and related spaces, including non-singular toric varieties, toric manifolds, intersections of quadrics and…

We study finitely generated nilpotent groups $G$ given by full rank finite presentations $\langle A \mid R\rangle$ in the variety $\mathcal{N}_c$ of nilpotent groups of class at most $c$, where $c \geq 2$. We prove that if the deficiency…

We classify torsion actions of free wreath products of arbitrary compact quantum groups and use this to prove that if $\mathbb{G}$ is a torsion-free compact quantum group satisfying the strong Baum-Connes property, then…

A subset S of a group G invariably generates G if G = <s^(g(s)) | s in S> for each choice of g(s) in G, s in S. In this paper we study invariable generation of infinite groups, with emphasis on linear groups. Our main result shows that a…

In this work we investigate tensor completions of groups by associative rings, which were introduced by R.Lyndon and G.Baumslag in 1960s. The main result states that there exists an algorithm that decides if a given finite system of…

A conjecture of Roseberger asserts that every generalised triangle group either is virtually soluble or contains a non-abelian free subgroup. Modulo two exceptional cases, we verify this conjecture for generalised triangle groups of type…

We obtain a forcing construction that shows that it is consistent that the torsion-free Abelian group $\mathbb{Q}^{(\lambda)}$ admits a Hausdorff group topology which is also $\mathcal{U}$-compact and contains no non-trivial convergent…

The first example of a non-residually finite group in the classes of finitely presented small-cancelation groups, automatic groups, and CAT(0) groups was constructed by Wise as the fundamental group of a complete square complex (CSC for…

A rank $n$ generalized Baumslag-Solitar group is a group that splits as a finite graph of groups such that all vertex and edge groups are isomorphic to $\mathbb{Z}^n$. In this paper we classify these groups in terms of their separability…

We show that the finite simply connected 2-complexes of nonpositive planar sectional curvature are collapsible. Moreover, we show that each finite connected 2-complex with negative planar sectional curvature and fundamental group…

We classify the groups quasi-isometric to a group generated by finite-order elements within the class of one-ended hyperbolic groups which are not Fuchsian and whose JSJ decomposition over two-ended subgroups does not contain rigid vertex…

Using the concept of algebraically closed groups, we prove that there is a countable torsion free group with exactly two conjugacy classes.

In recent years, knapsack problems for (in general non-commutative) groups have attracted attention. In this paper, the knapsack problem for wreath products is studied. It turns out that decidability of knapsack is not preserved under…

We show that any nonabelian free group $F$ of finite rank is homogeneous; that is for any tuples $\bar a$, $\bar b \in F^n$, having the same complete $n$-type, there exists an automorphism of $F$ which sends $\bar a$ to $\bar b$. We further…

We describe various classes of infinitely presented groups that are condensation points in the space of marked groups. A well-known class of such groups consists of finitely generated groups admitting an infinite minimal presentation. We…