Related papers: Splitting Line Patterns in Free Groups

A JSJ decomposition of a group is a splitting that allows one to classify all possible splittings of the group over a certain family of edge groups. Although JSJ decompositions are not unique in general, Guirardel--Levitt have constructed a…

We study line patterns in a free group by considering the topology of the decomposition space, a quotient of the boundary at infinity of the free group related to the line pattern. We show that the group of quasi-isometries preserving a…

We introduce the notion of a conjugation-free geometric presentation for a fundamental group of a line arrangement's complement, and we show that the fundamental groups of the following family of arrangements have a conjugation-free…

We characterize the ``best'' model geometries for the class of virtually free groups, and we show that there is a countable infinity of distinct ``best'' model geometries in an appropriate sense--these are the maximally symmetric trees. The…

We prove that a subset of a virtually free group is rational if and only if the language of geodesic words representing its elements (in any generating set) is rational and that the language of geodesics representing conjugates of elements…

We study the quasi-isometric rigidity of a large family of finitely generated groups that split as graphs of groups with virtually free vertex groups and two-ended edge groups. Let $G$ be a group that is one-ended, hyperbolic relative to…

Let $G$ be a hyperbolic group that splits as a graph of free groups with cyclic edge groups. We prove that, unless $G$ is isomorphic to a free product of free and surface groups, every finite abelian group $M$ appears as a direct summand in…

Using the canonical JSJ splitting, we describe the outer automorphism group $\Out(G)$ of a one-ended word hyperbolic group $G$. In particular, we discuss to what extent $\Out(G)$ is virtually a direct product of mapping class groups and a…

This paper deals with graph automaton groups associated with trees and some generalizations. We start by showing some algebraic properties of tree automaton groups. Then we characterize the associated semigroup, proving that it is…

We introduce the concept of a pants decomposition for a finitely generated free group and construct the corresponding pants graph. A pants decomposition of a free group leads to the formation of a simplicial graph, referred to as the pants…

The topology of the Bowditch boundary of a relatively hyperbolic group pair gives information about relative splittings of the group. It is therefore interesting to ask if there is generic behavior of this boundary. The purpose of this…

We classify all possible JSJ decompositions of doubles of free groups of rank two and we then compute the Makanin-Razborov diagram of a particular double of a free group and deduce that in general limit groups are not freely subgroup…

A group G is called subgroup conjugacy separable (abbreviated as SCS), if any two finitely generated and non-conjugate subgroups of G remain non-conjugate in some finite quotient of G. We prove that the free groups and the fundamental…

A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We construct the JSJ decomposition of a vGBS group over abelian groups. We prove that…

A group G is a vGBS group if it admits a decomposition as a finite graph of groups with all edge and vertex groups finitely generated and free abelian. We prove that the multiple conjugacy problem is solvable between two n-tuples A and B of…

We present the results of computer experiments suggesting that the probability that a random multiword in a free group is virtually geometric decays to zero exponentially quickly in the length of the multiword. We then prove this fact.

A group $G$ is called subgroup conjugacy separable if for every pair of non-conjugate finitely generated subgroups of $G$, there exists a finite quotient of $G$ where the images of these subgroups are not conjugate. We prove that limit…

Let $G$ be a finite group, and let $\Delta(G)$ be the prime graph built on the set of conjugacy class sizes of $G$: this is the simple undirected graph whose vertices are the prime numbers dividing some conjugacy class size of $G$, two…

We show that all finitely generated free-by-cyclic groups are conjugacy separable: if a finitely generated group $G$ surjects onto $\mathbb{Z}$ with free kernel, then for every pair of non-conjugate elements $g,h\in G$, there exists a…

Generalized Baumslag-Solitar groups are defined as fundamental groups of graphs of groups with infinite cyclic vertex and edge groups. Forester proved (in "On uniqueness of JSJ decompositions of finitely generated groups", Comment. Math.…