Related papers: Endomorphisms of free Steiner quasigroups
Automatic groups admitting prefix closed automatic structures with uniqueness are characterized as the quotients of free groups by normal subgroups possessing sets of free generators satisfying certain language-theoretic conditions.
Given a Coxeter system with a fixed Coxeter element, there is a surjective group morphism $\Psi$ from the standard to the dual Artin groups. We give conditions that are sufficient, necessary or equivalent to $\Psi$ being an isomorphism. In…
A combinatorial summation identity over the lattice of labelled hypertrees is established that allows one to gain concrete information on the Euler characteristics of various automorphism groups of free products of groups.
In this paper we characterize when a Cayley automaton semigroup is a group, is trivial, is finite, is free, is a left zero semigroup, or is a right zero semigroup.
Free products of semisimple tesnor categories are constructed with the help of polygonal presentation. The semisimplicity criterion is obtained for the Bisch-Jones' planar algebras as a byproduct.
A random graph of free groups contains a surface subgroup
The Baer-Specker group is the product of countably many copies of the additive group Z of integers. Assuming the continuum hypothesis, we construct a pure subgroup G of the Baer-Specker group with the following properties. Every…
Homomorphism indistinguishability is a way of characterising many natural equivalence relations on graphs. Two graphs $G$ and $H$ are called homomorphism indistinguishable over a graph class $\mathcal{F}$ if for each $F \in \mathcal{F}$,…
We investigate the "two-parameter" quantum symmetry groups that we previously constructed with Skalski, with the conclusion that some of these quantum groups, namely those without singletons, are "super-easy" in a suitable sense, that we…
We define and develop the notion of a discretisable quasi-action. It is shown that a cobounded quasi-action on a proper non-elementary hyperbolic space $X$ not fixing a point of $\partial X$ is quasi-conjugate to an isometric action on…
We prove that the boundary dynamics of the (semi)group generated by the enriched dual transducer characterizes the algebraic property of being free for an automaton group. We specialize this result to the class of bireversible transducers…
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…
The construction of bases for quotients is an important problem. In this paper, applying the method of rewriting systems, we give a unified approach to construct sections---an alternative name for bases in semigroup theory---for quotients…
Finite covers are a technique for building new structures from simpler ones. The original motivation to study finite covers is in the Ladder theorem of Zilber which describes how totally categorical structures are built from strictly…
We show that every periodic element of the free idempotent generated semigroup on an arbitrary biordered set belongs to a subgroup of the semigroup.
An endomorphism of a free group is called primitivity preserving if it takes every primitive element to another primitive. In this paper we prove that every primitivity preserving endomorphism of a free group of a finite rank n > 2 is an…
Working in the soft-element (classical) viewpoint, we introduce \emph{soft bitopological groups}: soft groups endowed with two soft topologies such that the induced topologies on the set of soft elements make the soft-element group into a…
We show that every rank two $p$-group acts freely and smoothly on a product of two spheres. This follows from a more general construction: given a smooth action of a finite group $G$ on a manifold $M$, we construct a smooth free action on…
In this note we give a new proof of the fact that an elementary subgroup (in the sense of first-order theory) of a non abelian free group $\mathbb{F}$ must be a free factor. The proof is based on definability of orbits of elements of under…
Factors $\frac{X}{Y}$ in a free group $F$ with $Y$ normal in $X$ are considered. Precise results on the free structure of ${Y}$ relative to the free structure of ${X}$ when $\frac{X}{Y}$ is abelian are obtained. Some extensions and…