Quasi-isometric embedding from the generalised Thompson's group TnT_n to TT

Brown has defined the generalised Thompson's group $F_n$, $T_n$, where $n$ is an integer at least $2$ and Thompson's groups $F= F_2$ and $T =T_2$ in the 80's. Burillo, Cleary and Stein have found that there is a quasi-isometric embedding from $F_n$ to $F_m$ where $n$ and $m$ are positive integers at least 2. We show that there is a quasi-isometric embedding from $T_n$ to $T_2$ for any $n \geq 2$ and no embeddings from $T_2$ to $T_n$ for $n \geq 3$

Combinatorial and metric properties of Thompson's group T

We discuss metric and combinatorial properties of Thompson's group T, such as the normal forms for elements and uniqueness of tree pair diagrams. We relate these properties to those of Thompson's group F when possible, and highlight combinatorial differences between the two groups. We define a set of unique normal forms for elements of T arising from minimal factorizations of elements into convenient pieces. We show that the number of carets in a reduced representative of T estimates the word length, that F is undistorted in T, and that cyclic subgroups of T are undistorted. We show that every element of T has a power which is conjugate to an element of F and describe how to recognize torsion elements in T

Geometric Topology

Geometric topology has seen significant advances in the understanding and application of infinite symmetries and of the principles behind them. On the one hand, for advances in (geometric) group theory, tools from algebraic topology are applied and extended; on the other hand, spectacular results in topology (e.g., the proofs of new cases of the Novikov conjecture or the Atiyah conjecture) were only possible through a combination of methods of homotopy theory and new insights in the geometry of groups. This workshop focused on the rich interplay between algebraic topology and geometric group theory

A Geometric Study of Commutator Subgroups

Let G be a group and G' its commutator subgroup. Commutator length (cl) and stable commutator length (scl) are naturally defined concepts for elements of G'. We study cl and scl for two classes of groups. First, we compute scl in generalized Thompson's groups and their central extensions. As a consequence, we find examples of finitely presented groups in which scl takes irrational (in fact, transcendental) values. Second, we study large scale geometry of the Cayley graph of a commutator subgroup with respect to the canonical generating set of all commutators. When G is a non-elementary hyperbolic group, we prove that, for any n, there exists a quasi-isometrically embedded, dimension n integral lattice in this graph. Thus this graph is not hyperbolic, has infinite asymptotic dimension, and has only one end. For a general finitely presented group, we show that this graph is large scale simply connected

Tree automorphisms and quasi-isometries of Thompson's group F

We prove that automorphisms of the infinite binary rooted tree T2 do not yield quasi-isometries of Thompson's group F, except for the map which reverses orientation on the unit interval, a natural outer automorphism of F. This map, together with the identity map, forms a subgroup of Aut(T2) consisting of 2-adic automorphisms, following standard terminology used in the study of branch groups. However, for more general p, we show that the analgous groups of p-adic tree automorphisms do not give rise to quasiisometries of F(p)

Random subgroups of Thompson's group FF

We consider random subgroups of Thompson's group $F$ with respect to two natural stratifications of the set of all $k$ generator subgroups. We find that the isomorphism classes of subgroups which occur with positive density are not the same for the two stratifications. We give the first known examples of {\em persistent} subgroups, whose isomorphism classes occur with positive density within the set of $k$-generator subgroups, for all sufficiently large $k$. Additionally, Thompson's group provides the first example of a group without a generic isomorphism class of subgroup. Elements of $F$ are represented uniquely by reduced pairs of finite rooted binary trees. We compute the asymptotic growth rate and a generating function for the number of reduced pairs of trees, which we show is D-finite and not algebraic. We then use the asymptotic growth to prove our density results.Comment: 37 pages, 11 figure

Elementary amenable subgroups of R. Thompson's group F

The subgroup structure of Thompson's group F is not yet fully understood. The group F is a subgroup of the group PL(I) of orientation preserving, piecewise linear self homeomorphisms of the unit interval and this larger group thus also has a poorly understood subgroup structure. It is reasonable to guess that F is the "only" subgroup of PL(I) that is not elementary amenable. In this paper, we explore the complexity of the elementary amenable subgroups of F in an attempt to understand the boundary between the elementary amenable subgroups and the non-elementary amenable. We construct an example of an elementary amenable subgroup up to class (height) omega squared, where omega is the first infinite ordinal.Comment: 20 page