José Burillo

Metrics and embeddings of generalizations of Thompson’s group

The distance from the origin in the word metric for generalizations F (p) of Thompson’s group F is quasi-isometric to the number of carets in the reduced rooted tree diagrams representing the elements of F (p). This interpretation of the metric is used to prove that every F (p) admits a quasiisometric embedding into every F (q), and also to study the behavior of the shift maps under these embeddings. The first example of a finitely presented, infinite, simple group was discovered by R.J. Thompson in 1965. This group G arose in the study of logic and it has many interesting descriptions. In 1974, Higman [10] used a description as the group of automorphisms of the Jonsson-Tarski algebra to generalize G to an infinite family of groups. G has another description as a group of homeomorphisms of the Cantor set. It has subgroups which may be described as homeomorphism groups of different objects, for instance, the circle and the unit interval. Among the people studying these generalizations are Brown [3] and Bieri and Strebel, in a set of unpublished notes [1]. In this paper we will concentrate on the groups F (p), which correspond to the groups Fp,∞ in [3]. These families of groups have also been extensively studied by Brin and Guzman [2]. The group F (2), commonly known as Thompson’s group F , can be regarded as the group of piecewise-linear, orientation-preserving homeomorphisms of the interval [0, 1] which have breakpoints only at dyadic points and whose slopes, where defined, are powers of two. In 1984 Brown and Geoghegan [4] found it to be the first example of a finitely presented, infinite-dimensional, torsion-free FP∞ group. This fact has been extended to all F (p) in [3], and studied by Stein [11], where generalizations to more general groups of homeomorphisms with general rational breakpoint sets are considered. Cleary [7] has studied these properties for groups of piecewise-linear homeomorphisms with irrational breakpoints and slopes. In [5] Burillo established the relationship between the word metric of Thompson’s group F and an estimate of the distance derived from the unique normal form of the elements. This algebraic estimate is quasi-isometric to the word metric and was used to prove that some subgroups are nondistorted in F . In this paper we find a geometric estimate of the word metric in terms of rooted tree diagrams (see [6]), show that this is quasi-isometric to the word metric, and use this interpretation to prove that some embeddings of F (p) into F (q), including some of those outlined in [2], are quasi-isometric. The interpretations of these embeddings in terms of Received by the editors September 25, 1998 and, in revised form, August 11, 1999. 2000 Mathematics Subject Classification. Primary 20F65; Secondary 20F05, 20F38, 20E99, 05C25. c ©2000 American Mathematical Society

ON GROUPS WHOSE GEODESIC GROWTH IS POLYNOMIAL

This paper records some observations concerning geodesic growth functions. If a nilpotent group is not virtually cyclic then it has exponential geodesic growth with respect to all finite generating sets. On the other hand, if a finitely generated group G has an element whose normal closure is abelian and of finite index, then G has a finite generating set with respect to which the geodesic growth is polynomial (this includes all virtually cyclic groups).

Growth of Positive Words in Thompson's Group F

Abstract Although it is well known that the growth of Thompsons group F is exponential, the exact growth function is still unknown. Elements of its submonoid of positive words can be described using a binary rooted tree, whose norm can be computed assigning weights to each caret. Combining this fact with a combinatorial argument, the growth function of the submonoid is computed and thus providing a first step in the computation of the growth function of the group, as well as a lower bound for the growth rate for the group.

Computational Explorations in Thompson’s Group F

Here 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 question whether or not Thompson’s group is amenable. We also describe experiments to estimate the exponential growth rate of F and the rate of escape of symmetric random walks with respect to the standard generating set.

COMMENSURATIONS AND SUBGROUPS OF FINITE INDEX OF THOMPSON'S GROUP F

We determine the abstract commensurator Com.F/ of Thompson’s group F and describe it in terms of piecewise linear homeomorphisms of the real line. We show Com.F/ is not finitely generated and determine which subgroups of finite index in F are isomorphic to F . We also show that the natural map from the commensurator group to the quasi-isometry group of F is injective. 20E34, 20F65; 26A30, 20F28

Commensurations and metric properties of Houghton’s groups

We describe the automorphism groups and the abstract commensurators of Houghtons groups. Then we give sharp estimates for the word metric of these groups and deduce that the commensurators embed into the corresponding quasi-isometry groups. As a further consequence, we obtain that the Houghton group on two rays is at least quadratically distorted in those with three or more rays.

Conjugacy in Houghton's groups

Let n ∈ N. Houghton’s group Hn is the group of permutations of {1, . . . , n} × N, that eventually act as a translation in each copy of N. We prove the solvability of the conjugacy problem and conjugator search problem for Hn, n ≥ 2.

Metric properties of Baumslag–Solitar groups

We compute estimates for the word metric of Baumslag–Solitar groups in terms of the Brittons lemma normal form. As a corollary, we find lower bounds for the growth rate for the groups BS(p, q) with 1 < p ≤ q.

Growth of positive words and lower bounds of the growth rate for Thompson’s groups F(p)

Let F(p), p ≥ 2 be the family of generalized Thompson’s groups. Here, F(2) is the famous Richard Thompson’s group usually denoted by F. We find the growth rate of the monoid of positive words in F(p) and show that it does not exceed p + 1/2. Also, we describe new normal forms for elements of F(p) and, using these forms, we find a lower bound for the growth rate of F(p) in its natural generators. This lower bound asymptotically equals (p − 1/2)log2e + 1/2 for large values of p.

The automorphism group of Thompson's group F: subgroups and metric properties.

We describe some of the geometric properties of the automorphism group Aut(F) of Thompsons group F. We give realizations of Aut(F) geometrically via periodic tree pair diagrams, which lead to natural presentations and give effective methods for estimating the word length of elements. We study some natural subgroups of Aut(F) and their metric properties. In particular, we show that the subgroup of inner automorphisms of F is at least quadratically distorted in Aut(F), whereas other subgroups of Aut(F) isomorphic to F are undistorted.