THOMPSON'S GROUP F
By James M. Belk. 2007
104 citations recorded by the source corpus.
We introduce forest diagrams and strand diagrams for elements of Thompson's group F. A forest diagram is a pair of infinite, bounded binary forests together with an order-preserving bijection of the leaves. Using forest diagrams, we derive a simple length formula for elements of F, and we discuss applications to the geometry of the Cayley graph, including a new upper bound on the isoperimetric constant (a.k.a. Cheeger constant) of F. Strand diagrams are similar to tree diagrams, but they can be concatenated like braids. Motivated by the fact that configuration spaces are classifying spaces for braid groups, we present a classifying space for F that is the ``configuration space'' of finitely many points on a line, with the points allowed to split and merge in pairs. Strand diagrams are related to a description of F as a groupoid, which we use to derive presentations for F, T, V, and the braided Thompson group BV. In addition to the new results, we include a thorough exposition of the basic theory of the group F. Highlights include a simplified proof that the commutator subgroup of F is simple, a discussion of open problems (with a focus on amenability), and a simplified derivation of the standard presentation and normal forms for F using forest diagrams.
View Paper: https://www.semanticscholar.org/paper/b09493169779a50ad1bf1d1721e6bc8c33b44845
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
- Yash Lodha
- Justin Tatch Moore
- R. Strebel
- R. Bieri
- Andrés Navas
- Ilijas Farah
- Marion Scheepers
Citations in the corpus
- On Groups of PL-homeomorphisms of the Real Line (2014, background): “The papers [BW07] and [BCR08] study finite index subgroups of Thompson’s group F = G([0, 1];Z[1/2], gp(2)).”
- A nonamenable finitely presented group of piecewise projective homeomorphisms (2016, background): “At the heart of Banach and Tarski’s paradoxical decomposition of the sphere is the existence of a paradoxical decomposition of the 1 is research was supported in part by NSF grant DMS–1262019.”