Related papers: Cyclically presented groups as Labelled Oriented G…

A cyclic presentation of a group is a presentation with an equal number of generators and relators that admits a particular cyclic symmetry. We characterise the orientable, non-orientable, and redundant cyclic presentations and obtain…

A mixed dihedral group is a group $H$ with two disjoint subgroups $X$ and $Y$, each elementary abelian of order $2^n$, such that $H$ is generated by $X\cup Y$, and $H/H'\cong X\times Y$. In this paper, for each $n\geq 2$, we construct a…

This paper concerns finite groups of class (at most) two and of odd prime exponent $p$. Such a group is called special if the center lies within its derived group. Every group of class 2 and exponent $p$ can be uniquely expressed as the…

We show that the \s{\phi}-labeled Thompson groups and the twisted Brin--Thompson groups are boundedly acyclic. This allows us to prove several new embedding results for groups. First, every group of type $F_n$ embeds quasi-isometrically…

We present novel constructions concerning the homology of finitely generated groups. Each construction draws on ideas of Gilbert Baumslag. There is a finitely presented acyclic group $U$ such that $U$ has no proper subgroups of finite index…

We give a simpler proof using automata theory of a recent result of Kapovich, Weidmann and Myasnikov according to which so-called benign graphs of groups preserve decidability of the generalized word problem. These include graphs of groups…

Associated to an abelian variety over a number field are several interesting and related groups: the motivic Galois group, the Mumford-Tate group, $\ell$-adic monodromy groups, and the Sato-Tate group. Assuming the Mumford-Tate conjecture,…

We show that if G is any nilpotent, finite group, and the commutator subgroup of G is cyclic, then every connected Cayley graph on G has a hamiltonian cycle.

A result of Baumslag and Roseblade states that a finitely presented subgroup of the direct product of two free groups is virtually a direct product of free groups. In this paper we generalise this result to the class of cyclic subgroup…

We prove that all finitely generated fully residually free groups (limit groups) have a sequence of finite dimensional unitary representations that `strongly converge' to the regular representation of the group. The corresponding statement…

We establish a bijective correspondence between Smirnov words with balanced letter multiplicities and Hamiltonian paths in complete $m$-partite graphs $K_{n,n,\ldots,n}$. This bijection allows us to derive closed inclusion-exclusion…

We prove that a random Cayley graph on a group of order $N$ has clique number $O(\log N \log \log N)$ with high probability. This bound is best possible up to the constant factor for certain groups, including~$\mathbb{F}_2^n$, and improves…

Let $M$ be a closed, connected, orientable topological four-manifold with $H_1(M)$ nontrivial and free abelian, $b_2(M)\ne 0, 2$, and $\chi(M)\ne 0$. We show that if $G$ is a finite group of 2-rank $\le 1$ which admits a homologically…

To reinforce the analogy between the mapping class group and the Cremona group of rank $2$ over an algebraic closed field, we look for a graph analoguous to the curve graph and such that the Cremona group acts on it non-trivially. A…

Let $G$ be a Polish (i.e., complete separable metric topological) group. Define $G$ to be an algebraically determined Polish group if for any Polish group $L$ and algebraic isomorphism $\varphi: L \mapsto G$, we have that $\varphi$ is a…

We consider the class of finitely generated groups which have a normal form computable in logspace. We prove that the class of such groups is closed under finite extensions, finite index subgroups, direct products, wreath products, and also…

We enumerate certain geometric equivalence classes of subgraphs induced by Hamiltonian paths and cycles in complete graphs. Said classes are orbits under the action of certain direct products of dihedral and cyclic groups on sets of strings…

We determine all factorisations $X=AB$, where $X$ is a finite almost simple group and $A,B$ are core-free subgroups such that $A\cap B$ is cyclic or dihedral. As a main application, we classify the graphs $\Gamma$ admitting an almost simple…

Building on previous results concerning hyperbolicity of groups of Fibonacci type, we give an almost complete classification of the (non-elementary) hyperbolic groups within this class. We are unable to determine the hyperbolicity status of…

The following refinement of the Higman embedding theorem is proved: A finitely generated group $R$ is recursively presented if and only if there exists a quasi-isometric malnormal embedding of $R$ into a finitely presented group $H$ such…