Related papers: Branch Groups
This article is a survey on the braid groups, the Artin groups, and the Garside groups. It is a presentation, accessible to non-experts, of various topological and algebraic aspects of these groups. It is also a report on three points of…
Similar pictures appear in various branches of mathematics. Sometimes this similarity gives rise to deep theorems. Mentioning such a similarity between hexagonal tilings, cubes in 3-space, configurations of lines and braid groups, we prove…
We study the class of groups generated by automata that act essentially freely on the boundary of a rooted tree. In the process we establish and discuss some general tools for determining if a group belongs to this class, and explore the…
In this paper we elaborate on the structure of the semigroup tree and the regularities on the number of descendants of each node observed earlier. These regularites admit two different types of behavior and in this work we investigate which…
The aim of this article is to discuss and clarify the notion of fractality for subgroups of the group of automorphisms of a regular rooted tree. For this purpose we define three types of fractality. We show that they are not equivalent, by…
We study an abstract notion of tree structure which lies at the common core of various tree-like discrete structures commonly used in combinatorics: trees in graphs, order trees, nested subsets of a set, tree-decompositions of graphs and…
We develop a notion of groups that act acylindrically and non-elementarily on simplicial trees, which we call acylindrically arboreal groups. We then prove a complete classification of when graph products of groups and the fundamental…
We give a structural description of the normal subgroups of subgroups of finite index in branch groups in terms of rigid stabilizers. This gives further insight into the structure lattices of branch groups introduced by the second author.…
We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on reducing the problem to showing amenability…
Initial steps are presented towards understanding which finitely generated groups are almost surely generated as semigroups by the path of a random walk on the group.
Latent tree analysis seeks to model the correlations among a set of random variables using a tree of latent variables. It was proposed as an improvement to latent class analysis --- a method widely used in social sciences and medicine to…
This expository article introduces the topic of roots in a compact Lie group. Compared to the many other treatments of this standard topic, I intended for mine to be relatively elementary, example-driven, and free of unnecessary…
This is a survey paper based on my talk at the Workshop on Orbifolds and String Theory, the goal of which was to explain the role of groupoids and their classifying spaces as a foundation for the theory of orbifolds.
The primary tool for analysing groups acting on trees is Bass--Serre Theory. It is comprised of two parts: a decomposition result, in which an action is decomposed via a graph of groups, and a construction result, in which graphs of groups…
Rooted phylogenetic networks are used by biologists to infer and represent complex evolutionary relationships between species that cannot be accurately explained by a phylogenetic tree. Tree-child networks are a particular class of rooted…
On the transversals of a subgroup of a group, using the binary operation of the group, structural mappings are defined. Based on these mappings, the notion of the hypergroup over the group is introduced, which generalizes the notion of the…
We consider groups that act on spherically symmetric rooted trees and study the associated representation of the group on the space of locally constant functions on the boundary of the tree. We introduce and discuss the new notion of…
How does an irreducible representation of a group behave when restricted to a subgroup? This is part of branching problems, which are one of the fundamental problems in representation theory, and also interact naturally with other fields of…
This is a survey of results on random group presentations, and on random subgroups of certain fixed groups. Being a survey, this paper does not contain new results, but it offers a synthetic view of a part of this very active field of…
Bass-Serre theory provides a powerful framework for studying group actions on trees. While extremely effective for structural questions in group theory, it is less suited to the systematic construction of group actions with prescribed local…