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We report on recent progress concerning the relationship that exists between the algebraic structure of a finite group and certain features of its class-size prime graph.
This paper is the second part of a two-part study of an elementary functorial construction of tesselated surfaces from finite groups. This elementary construction was discussed in the first part and generally results in a large collection…
Topological full groups originated from the theory of topological dynamical systems and have been having considerable impact on group theory in recent years. This text represents an introduction/survey on topological full groups. After…
We describe the action of the mapping class group $M(g,n)$ on the fundamental group of $T_{g,n}$, a compact orientable topological surface of positive genus $g$ with $n$ marked points. This is achieved by computing the image of the…
The work of Mann and Rafi gives a classification surfaces $\Sigma$ when $\textrm{Map}(\Sigma)$ is globally CB, locally CB, and CB generated under the technical assumption of tameness. In this article, we restrict our study to the pure…
The goal of this paper is to describe a theoretical construction of an infinite collection of non-classical Schottky groups. We first show that there are infinitely many non-classical noded Schottky groups on the boundary of Schottky space,…
Rational conformal field theories produce a tower of finite-dimensional representations of surface mapping class groups, acting on the conformal blocks of the theory. We review this formalism. We show that many recent mathematical…
We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides…
Let $S(n)$ be the infinite-type surface with infinite genus and $n \in \mathbb{N}$ ends, all of which are accumulated by genus. The mapping class group of this surface, $\mod(S(n))$, is a Polish group that is not countably generated, but it…
We study the mapping class group of a nontrivial irreducible shift of finite type: the group of flow equivalences of its mapping torus modulo isotopy. This group plays for flow equivalence the role that the automorphism group plays for…
We exhibit a finitely generated group $\M$ whose rational homology is isomorphic to the rational stable homology of the mapping class group. It is defined as a mapping class group associated to a surface $\su$ of infinite genus, and…
We define and study analogs of curve graphs for infinite type surfaces. Our definitions use the geometry of a fixed surface and vertices of our graphs are infinite multicurves which are bounded in both a geometric and a topological sense.…
We show that the natural map from the mapping class groups of surfaces to the automorphism groups of free groups, induces an infinite loop map on the classifying spaces of the stable groups after plus construction. The proof uses…
A generating function of the number of homomorphisms from the fundamental group of a compact oriented or non-orientable surface without boundary into a finite group is obtained in terms of an integral over a real group algebra. We calculate…
The mapping class group of a surface with one boundary component admits numerous interesting representations including as a group of automorphisms of a free group and as a group of symplectic transformations. Insofar as the mapping class…
We prove that the mapping class group of the one-holed Cantor tree surface is acyclic. This in turn determines the homology of the mapping class group of the once-punctured Cantor tree surface (i.e. the plane minus a Cantor set), in…
Recently there has been a lot of research and progress in profinite groups. We survey some of the new results and discuss open problems. A central theme is decompositions of finite groups into bounded products of subsets of various kinds…
We consider the problem of realizing a group as the fundamental group of a graph of groups where the vertex groups are restricted to certain classes (for example, coming from a certain finite list of groups, or having bounded geometric…
We develop a theory of large scale geometry of metrisable topological groups that, in a significant number of cases, allows one to define and identify a unique quasi-isometry type intrinsic to the topological group. Moreover, this…
We study the action of (big) mapping class groups on the first homology of the corresponding surface. We give a precise characterization of the image of the induced homology representation.