Related papers: On joins and intersections of subgroups in free gr…
We discuss a partial normalisation of a finite graph of finite groups $(\Gamma(-), X)$ which leaves invariant the fundamental group. In conjunction with an easy graph-theoretic result, this provides a flexible and rather useful tool in the…
We establish a combination theorem for parafree groups. These groups were introduced by Baumslag in the sixties. One of the current motivations for a better understanding of their structure is that they show up naturally in connection with…
We enumerate the independent sets of several classes of regular and almost regular graphs and compute the corresponding generating functions. We also note the relations between these graphs and other combinatorial objects and, in some…
We discuss in the context of finite extensions two classical theorems of Takahasi and Howson on subgroups of free groups. We provide bounds for the rank of the intersection of subgroups within classes of groups such as virtually free…
We classify modules and rings with some specific properties of their intersection graphs. In particular, we describe rings with infinite intersection graphs containing maximal left ideals of finite degree. This answers a question raised in…
We investigate groups whose Cayley graphs have poor\-ly connected subgraphs. We prove that a finitely generated group has bounded separation in the sense of Benjamini--Schramm--Tim\'ar if and only if it is virtually free. We then prove a…
A finitely generated group admits a decomposition, called its Grushko decomposition, into a free product of freely indecomposable groups. There is an algorithm to construct the Grushko decomposition of a finite graph of finite rank free…
We show that the level sets of automorphisms of free groups with respect to the Lipschitz metric are connected as subsets of Culler-Vogtmann space. In fact we prove our result in a more general setting of deformation spaces. As…
We show that in the free group of rank 3, given an arbitrary number of automorphisms, the intersection of their fixed subgroups is equal to the fixed subgroup of some other single automorphism.
We develop a categorical framework for studying graphs of groups and their morphisms, with emphasis on pullbacks. More precisely, building on classical work by Serre and Bass, we give an explicit construction of the so-called…
The intersection ideal graph $\Gamma(S)$ of a semigroup $S$ is a simple undirected graph whose vertices are all nontrivial left ideals of $S$ and two distinct left ideals $I, J$ are adjacent if and only if their intersection is nontrivial.…
Geometric semigroup theory is the systematic investigation of finitely-generated semigroups using the topology and geometry of their associated automata. In this article we show how a number of easily-defined expansions on finite semigroups…
A random graph of free groups contains a surface subgroup
A paradigm that was successfully applied in the study of both pure and algorithmic problems in graph theory can be colloquially summarized as stating that "any graph is close to being the disjoint union of expanders". Our goal in this paper…
Motivated by the theory of graph limits, we introduce and study the convergence and limits of linear representations of finite groups over finite fields. The limit objects are infinite dimensional representations of free groups in…
Let $G$ be a finite abelian group, written additively, and $H$ a subgroup of~$G$. The \emph{subgroup sum graph} $\Gamma_{G,H}$ is the graph with vertex set $G$, in which two distinct vertices $x$ and $y$ are joined if $x+y\in…
The theory of covering spaces is often used to prove the Nielsen-Schreier theorem, which states that every subgroup of a free group is free. We apply the more general theory of semicovering spaces to obtain analogous subgroup theorems for…
We give upper bounds, linear in rank, to the topological dimensions of the Gromov boundaries of the intersection graph, the free factor graph and the cyclic splitting graph of a finitely generated free group.
The intersection growth of a group $G$ is the asymptotic behavior of the index of the intersection of all subgroups of $G$ with index at most $n$, and measures the Hausdorff dimension of $G$ in profinite metrics. We study intersection…
The end compactification |\Gamma| of the locally finite graph \Gamma is the union of the graph and its ends, endowed with a suitable topology. We show that \pi_1(|\Gamma|) embeds into a nonstandard free group with hyperfinitely many…