Related papers: Link quandles are residually finite

We prove that finitely presented residually free groups are subgroup conjugacy separable. Furthermore, if they are of type $FP_\infty$, then they are also subgroup conjugacy distinguished. Using a connection between conjugacy separability…

We show that the second bounded cohomology of the free product of racks and quandles is infinite-dimensional as a real vector space. This is similar to the case of groups. As a corollary, we show that the second bounded cohomology of the…

Let $G$ be a real Lie group and $\Gamma < G$ be a discrete subgroup of $G$. Is $\Gamma$ residually finite? This paper describes known positive and negative results then poses some questions whose answers will lead to a fairly complete…

A (left) quandle is connected if its left multiplication group acts transitively. In 2014, Eisermann introduced the concept of quandle coverings, corresponding to so-called constant quandle cocycles that form a subset of quandle cocycles. A…

We construct a finitely generated residually finite group $G$ with the property that every finite index subgroup of $G$ contains a subgroup isomorphic to Promislow's group. Hence $G$ does not have a finite index subgroup with the unique…

Quandles can be regarded as generalizations of symmetric spaces. In the study of symmetric spaces, the notion of flatness plays an important role. In this paper, we define the notion of flat quandles, by referring to the theory of…

To better understand the fundamental quandle of a knot or link, it can be useful to look at finite quotients of the quandle. One such quotient is the $n$-quandle (or, when $n=2$, the {\em involutory} quandle). Hoste and Shanahan \cite{HS2}…

We study a family of finitely generated residually finite groups. These groups are doubles $F_2*_H F_2$ of a rank-$2$ free group $F_2$ along an infinitely generated subgroup $H$. Varying $H$ yields uncountably many groups up to isomorphism.

We establish a general criterion for the finite presentability of subdirect products of groups and use this to characterize finitely presented residually free groups. We prove that, for all $n\in\mathbb{N}$, a residually free group is of…

The residual finiteness growth of a group quantifies how well approximated the group is by its finite quotients. In this paper, we construct groups with arbitrarily large residual finiteness growth. We also demonstrate a new relationship…

In this article we develop the theory of residually finite rationally $p$ (RFR$p$) groups, where $p$ is a prime. We first prove a series of results about the structure of finitely generated RFR$p$ groups (either for a single prime $p$, or…

We show that every virtually torsion-free subgroup of the outer automorphism group of a conjugacy separable relatively hyperbolic group is residually finite. As a direct consequence, we obtain that the outer automorphism group of a limit…

Let C be a class of groups. We give sufficient conditions ensuring that a free product of residually C groups is again residually C, and analogous conditions are given for locally embeddable into C groups. As a corollary, we obtain that the…

We establish a canonical correspondence between connected quandles and certain configurations in transitive groups, called quandle envelopes. This correspondence allows us to efficiently enumerate connected quandles of small orders, and…

We show that the virtual second Betti number of a finitely generated, residually free group $G$ is finite if and only if $G$ is either free, free abelian or the fundamental group of a closed surface. We also prove a similar statement in…

We show that Out(G) is residually finite if G is a one-ended group that is hyperbolic relative to virtually polycyclic subgroups. More generally, if G is one-ended and hyperbolic relative to proper residually finite subgroups, the group of…

In this paper we study the residual solvability of the generalized free product of finitely generated nilpotent groups. We show that these kinds of structures are often residually solvable.

This is a survey of two papers joint with A. Borisov and a paper joint with I. Spakulova. It is based on my lectures at the conference "Groups St. Andrews 2009", Bath (August 2009). We prove that almost all 1-related groups with at least 3…

In this article we study a class of central extensions of $\mathbb{Z}\wr\mathbb{Z}$, as first described by Hall. On the one hand, we consider groups of this type with cyclic centre, our construction yields a rich class of groups. In…

Sets with a self-distributive operation (in the sense of $(a \triangleleft b) \triangleleft c = (a \triangleleft c) \triangleleft (b \triangleleft c))$, in particular quandles, appear in knot and braid theories, Hopf algebra classification,…