# An Irrational-slope Thompson's Group

@article{Burillo2018AnIT, title={An Irrational-slope Thompson's Group}, author={J. Burillo and B. Nucinkis and Lawrence Reeves}, journal={arXiv: Group Theory}, year={2018} }

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 binary trees. We show that its commutator group is simple. Finally, inspired by the case of Thompson's group F, we define a unique normal form for the elements of the group and study the metric properties for the elements based on this normal form. As a corollary… Expand

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#### References

SHOWING 1-10 OF 17 REFERENCES

Combinatorial and metric properties of Thompson's group T

- Mathematics
- 2005

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… Expand

METRIC PROPERTIES OF HIGHER-DIMENSIONAL THOMPSON'S GROUPS

- Mathematics
- 2008

Higher-dimensional Thompson's groups nV are finitely presented groups that generalize dyadic self-maps of the unit interval to dyadic self-maps of n-dimensional unit cubes. We describe some of the… Expand

Growth of Positive Words in Thompson's Group F

- Mathematics
- 2004

Abstract Although it is well known that the growth of Thompson's group F is exponential, the exact growth function is still unknown. Elements of its submonoid of positive words can be described using… Expand

On Groups of Pl-homeomorphisms of the Real Line

- Mathematics
- 2016

Richard J. Thompson invented his group F in the 60s; it is a group full of surprises: it has a finite presentation with 2 generators and 2 relators, and a derived group that is simple; it admits a… Expand

Metrics and embeddings of generalizations of Thompson’s group $F$

- Mathematics
- 1998

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… Expand

THOMPSON'S GROUP IS DISTORTED IN THE THOMPSON-STEIN GROUPS

- Mathematics
- 2011

We show that the inclusion map of the generalized Thompson groups F(n i ) is exponentially distorted in the Thompson―Stein groups F(n 1 , ... , n k ) for k > 1. One consequence is that F is… Expand

Quasi-Isometrically Embedded Subgroups of Thompson's GroupF

- Mathematics
- 1998

Abstract The goal of this paper is to study subgroups of Thompson's group F which are isomorphic to F × Z n and F × F . A result estimating the norm of an element of Thompson's group is found,… Expand

Cohomological and Metric Properties of Groups of Homeomorphisms of R

- Mathematics
- 2019

In recent years, the family of groups sharing features or design principles with classical Thompson groups has grown considerably. The workshop highlights new developments in this field with special… Expand

Regular subdivision in Z

- Illinois J. Math.,
- 2000

Minimal Length Elements of Thompson's Group F

- 2003

AbstractElements of the group are represented by pairs of binary trees and the structure of the trees gives insight into the properties of the elements of the group. The review section presents this… Expand