Related papers: Quasi-isometrically embedded subgroups of Thompson…

We introduce a new quasi-isometry invariant for finitely generated groups and show that every group with this property admits a subshift which is effectively closed by patterns and that cannot be realized as the topological factor of any…

A quasi-tree is a geodesic metric space quasi-isometric to a tree. We give a general construction of many actions of groups on quasi-trees. The groups we can handle include non-elementary (relatively) hyperbolic groups, rank 1 CAT(0)…

Recall that a group $G$ is said to be $\frac{3}{2}$-generated if every non-trivial element of $G$ belongs to a generating pair of $G$. Thompson's group $V$ was proved to be $\frac{3}{2}$-generated by Donoven and Harper in 2019. It was the…

We prove that for any finitely generated relatively hyperbolic group G and any symmetric endomorphism f of G with relatively quasiconvex image, Fixf is relatively quasiconvex subgroup of G.

In this paper, we present a notion of quasiconvexity in the setting of finitely-generated groups with hyperbolically embedded subgroups. Our main result shows that this notion yields uniform quasiconvex constants in the setting of coned-off…

We obtain a characterisation of confined subgroups of Thompson's group $F$. As a result, we deduce that orbital graph of a point under action of $F$ has uniformly subexponential growth if and only if this point is fixed by the commutator…

We show that for any metric space $M$ satisfying certain natural conditions, there is a finitely generated group $G$, an ultrafilter $\omega $, and an isometric embedding $\iota $ of $M$ to the asymptotic cone ${\rm Cone}_\omega (G)$ such…

We construct uncountably many finitely generated, pairwise non-isomorphic torsion-free groups, all of which fall into the same quasi-isometry class. This is done by considering Schur covering groups and group cohomology, with the necessary…

We show that Thompson's group F does not satisfy Cannon's almost convexity condition AC(n) for any integer n in the standard finite two generator presentation. To accomplish this, we construct a family of pairs of elements at distance n…

We describe the kernel of the canonical map from the Floyd boundary of a relatively hyperbolic group to its Bowditch boundary. Using our methods we then prove that a finitely generated group $H$ admitting a quasi-isometric map $\phi$ into a…

In his work on representations of Thompson's group $F$, Vaughan Jones defined and studied the $3$-\emph{colorable subgroup} $\mathcal{F}$ of $F$. Later, Ren showed that it is isomorphic with the Brown-Thompson group $F_4$. In this paper we…

We study the poset of hyperbolic structures on Thompson's group $F$ and its generalizations $F_n$ for $n \geq 2$. The global structure of this poset is as simple as one would expect, with the maximal non-elementary elements being two…

Let M be a smooth compact connected oriented manifold of dimension at least two endowed with a volume form. Assuming certain conditions on the fundamental group $\pi_1(M)$ we construct quasi-isometric embeddings of either free Abelian or…

The purpose of this paper is to study the properties of the irrational-slope Thompson's group $F_\tau$ introduced by Cleary in 1995. We construct presentations, both finite and infinite and we describe its combinatorial structure using…

This is a survey of our recent results on the amenability problem for Thompson's group $F$. They mostly concern esimating the density of finite subgraphs in Cayley graphs of $F$ for various systems of generators, and also equations in the…

We show that properties $F_n$ and $FP_n$ hold for a relatively hyperbolic group if and only if they hold for all the peripheral subgroups. As an application we show that there are at least countably many distinct quasi-isometry classes of…

In this paper a geometric characterization of the unique geodesics in Thompson's metric spaces is presented. This characterization is used to prove a variety of other geometric results. Firstly, it will be shown that there exists a unique…

We describe the results of some computational explorations in Thompson's group F. We describe experiments to estimate the cogrowth of F with respect to its standard finite generating set, designed to address the subtle and difficult…

Following a procedure due to V. Jones, using suitably normalized elements in a Temperley-Lieb-Jones (planar) algebra we introduce a 3-parametric family of unitary representations of the Thompson's group $F$ equipped with canonical (vacuum)…

We say A is a quasi-normal subgroup of the group G if the commensurator of A in G is all of G. We develop geometric versions of commensurators in finitely generated groups. In particular, g is an element of the commensurator of A in G iff…