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Confined subgroups of Thompson's group F and its embeddings into wobbling groups
[article]

2018
*
arXiv
*
pre-print

We obtain a characterisation

arXiv:1809.05146v2
fatcat:e74y7ajncnbkrf6rglyyxahsde
*of*confined subgroups*of**Thompson's**group**F*. ... This allows us to prove non-embeddability*of**F*into wobbling*groups**of**graphs*with uniformly subexponential growth. ... Is it true that for any non-confined subgroup H <*F**one*can find constant K such that for infinitely many n ∈ N there is a copy*of*a ball B*F*(n) in a ball*of**the*Schreier*graph*with radius Kn centered ...##
###
Thompson's groupFand uniformly finite homology

2009
*
Algebraic and Geometric Topology
*

This paper seeks to apply

doi:10.2140/agt.2009.9.2349
fatcat:bhan43s3vbev5hmjyttcmtn5wi
*the*results*of*Block and Weinberger to subgraphs*of**the**Cayley**graph**of**Thompson's**group**F*. ... This allows us to state*the*following proposition: Proposition 2.6 Let be*the*subgraph*of**the**Cayley**graph**of**Thompson's**group**F*(using*the*x 0 ; x 1 generating set) consisting*of*vertices in*the*positive ...##
###
Traveller Salesman Property and Richard Thompson's Group F
[article]

2005
*
arXiv
*
pre-print

*Thompson's*

*group*

*F*has TSP.

*The*aim

*of*this article is to disprove this conjecture. ... Recently Akhmedov introduced a

*property*

*of*finitely generated

*groups*called

*the*Traveller Salesman

*Property*(TSP). It was shown by Thurston that TSP implies non-amenability. ... In several conference talks, Akhmedov conjectured that

*Richard*

*Thompson's*

*group*

*F*has

*property*T S. If this were true, this would imply nonamenability

*of*

*F*. ...

##
###
Thompson's Group F and Uniformly Finite Homology
[article]

2009
*
arXiv
*
pre-print

This paper demonstrates

arXiv:0806.2877v2
fatcat:rqrzwu5tjzgxfi7v4fymxlp2yq
*the*uniformly finite homology developed by Block and Weinberger and its relationship to amenable spaces via applications to*the**Cayley**graph**of**Thompson's**Group**F*. ... This shows that if*F*is amenable, these subsets (which include every finitely generated submonoid*of**the*positive monoid*of**F*) must necessarily have measure zero. ...*Thompson's**Group**F**Thompson's**group**F*has been studied for several decades. ...##
###
On the density of Cayley graphs of R.Thompson's group F in symmetric generators
[article]

2019
*
arXiv
*
pre-print

It is known that a

arXiv:1909.01882v1
fatcat:nvkdodqfovhpxijtpo5gfaufuq
*group*with m generators is amenable iff*the*density*of**the*corresponding*Cayley**graph*equals 2m. A famous problem*on**the*amenability*of*R.*Thompson's**group**F*is still open. ... For an m-generated*group*,*the*density*of*its*Cayley**graph*in a given set*of*generators, is*the*supremum*of*densities taken over all its finite subgraphs. ...*The*proof is complete. Notice that in order to prove non-amenability*of**Thompson's**group**F*(if it is*the*fact), it suffices to find a kind*of*a doubling structure*on**the**Cayley**graph**of**the**group*. ...##
###
Evacuation schemes on Cayley graphs and non-amenability of groups
[article]

2021
*
arXiv
*
pre-print

In this paper we introduce a concept

arXiv:2112.09812v1
fatcat:vfg26sennfcitbu5mr6qdjv4t4
*of*an evacuation scheme*on**the**Cayley**graph**of*an infinite finitely generated*group*. ... We obtain a criterion for existing*of*such a scheme in terms*of*isoperimetric constant*of**the**graph*. We analyze R.*Thompson's**group**F*, for which*the*amenability*property*is a famous open problem. ...*Thompson's**group**F*by*the*following infinite*group*presentation x 0 , x 1 , x 2 , . . . | x j x i = x i x j+1 (i < j) . (1) This*group*was found by*Richard*J. Thompson in*the*60s. ...##
###
Page 4219 of Mathematical Reviews Vol. , Issue 98G
[page]

1998
*
Mathematical Reviews
*

R. (1-EMI; Ypsilanti, M1)
Introductory notes

*on**Richard**Thompson’s**groups*. Enseign. Math. (2) 42 (1996), no. 3-4, 215-256. ... Summary: “We study*the*behavior at infinity*of**the**Cayley**graph*I'(G, A)*of*a*group*G with respect to a finite generating set A after removing subsets quasi-isometric to*the*real line.” ...##
###
Some graphs related to Thompson's group F
[article]

2008
*
arXiv
*
pre-print

*The*Schreier

*graphs*

*of*

*Thompson's*

*group*

*F*with respect to

*the*stabilizer

*of*1/2 and generators x_0 and x_1, and

*of*its unitary representation in L_2([0,1]) induced by

*the*standard action

*on*

*the*interval ...

*The*induced subgraph

*of*

*the*right

*Cayley*

*graph*

*of*

*the*positive monoid

*of*

*F*containing all

*the*vertices

*of*

*the*form x_nv, where n>=0 and v is any word over

*the*alphabet x_0, x_1, is constructed. ... Introduction

*Thompson's*

*group*

*F*was discovered by

*Richard*Thompson in 1965. A lot

*of*fascinating

*properties*

*of*this

*group*were discovered later

*on*, many

*of*which are surveyed nicely in [CFP96] . ...

##
###
Systems of equations over the group ring of Thompson's group F
[article]

2022
*
arXiv
*
pre-print

We improve some results by Donnelly showing that there exist finite sets Y⊂

arXiv:2201.02308v1
fatcat:7dpjfjwtprfcrpyz6auktl2fli
*F*with*the**property*|AY| < 4/3|Y|, where A={x_0,x_1,x_2}. This implies some result*on**the*systems*of*equations. ... Recently Bartholdi proved*the*converse. As a consequence*of*this theorem, Kielak proved that R.*Thompson's**group**F*is amenable if and only if it satisfies*the*Ore condition. ... Let Γ = C(*F*, A) be*the*left*Cayley**graph**of**F*in generators x 0 , x 1 , x 2 . ...##
###
Thompson's Group F is not Minimally Almost Convex
[article]

2003
*
arXiv
*
pre-print

We prove that

arXiv:math/0301141v1
fatcat:5vz7olwwhzfnlkfjxasysratpe
*Richard**Thompson's**group**F*is not minimally almost convex with respect to*the*two standard generators. This improves upon a recent result*of*S. Cleary and J. Taback. ... These diagrams seem particularly well-suited for understanding*the**Cayley**graph*for*the*two standard generators. ... Specifically, we will use*the*length formula for forest diagrams to calculate distances in*the**Cayley**graph**of**F*. Let Γ denote*the**Cayley**graph**of**F*. ...##
###
Page 920 of Mathematical Reviews Vol. , Issue 2004b
[page]

2004
*
Mathematical Reviews
*

a certain kind

*of*subgraph*of**the**Cayley**graph*—to give a detailed descrip- tion*of*geodesics in*the**Cayley**graph**of**the*Brady-Bridson*group*G. ...*Richard**Thompson’s**group**F*, for example, is isomorphic to*the*diagram*group*D((x | x? = x), x) [see V. S. Guba and M. V. Sapir, Mem. Amer. Math. Soc. 130 (1997), no. 620, viii+117 pp.; MR 98f:20013]. ...##
###
Sub-dominant cogrowth behaviour and the viability of deciding amenability numerically
[article]

2016
*
arXiv
*
pre-print

We identify two potential sources

arXiv:1608.06703v2
fatcat:utsebdbtvretnhr6enpnpjljlu
*of*error. We then propose a modification*of**the*method that enables it to easily compute surprisingly accurate estimates for initial terms*of**the*cogrowth sequence. ... We critically analyse a recent numerical method due to*the*first author, Rechnitzer and van Rensburg, which attempts to detect amenability or non-amenability in a finitely generated*group*by numerically ... Acknowledgements*The*authors wish to thank Andrew Rechnitzer and Andrew Elvey-Price for helpful feedback*on*this work. ...##
###
Schreier graphs of actions of Thompson's group F on the unit interval and on the Cantor set
[article]

2014
*
arXiv
*
pre-print

Schreier

arXiv:1105.4017v3
fatcat:jdtzyltnvzfq5fbdsimqk3bzai
*graphs**of**the*actions*of**Thompson's**group**F**on**the*orbits*of*all points*of**the*unit interval and*of**the*Cantor set with respect to*the*standard generating set {x_0,x_1} are explicitly constructed ...*The*closure*of**the*space*of*pointed Schreier*graphs**of**the*action*of**F**on**the*orbits*of*dyadic rational numbers and corresponding Schreier dynamical system are described. ... Introduction*Thompson's**group**F*was first defined by*Richard*Thompson in 1965 in*the*context*of*his work*on*logic. ...##
###
Page 6715 of Mathematical Reviews Vol. , Issue 99j
[page]

1999
*
Mathematical Reviews
*

Polynomial upper bounds for

*the*Dehn function*of*R.*Thompson’s**group**F*. (English summary) J.*Group*Theory 1 (1998), no. 2, 203-211.*The*author shows that*the*well-known*group**F**of*R. ...*The*proof*of*this nice result uses*the*idea*of*quasiconvexity: a subgroup H*of*a*group*G is said to be quasiconvex if there exists € > 0 such that any geodesic in*the**Cayley**graph**of*G whose endpoints ...##
###
Sub-dominant Cogrowth Behavior and the Viability of Deciding Amenability Numerically

2017
*
Experimental Mathematics
*

In Section 6

doi:10.1080/10586458.2017.1353452
fatcat:qklu76z55bgdxl7i5ivmtkamci
*the*ERR method is modified to provide estimates*of*initial cogrowth values. Using this*the*first 2000 terms for*the*cogrowth function*of**Thompson's**group**F*are estimated. ... Kesten's criteria for amenability is given in terms*of**the*probability*of*a random walk*on**the**group*returning to its starting point. ... Acknowledgements*The*authors wish to thank Andrew Rechnitzer, Andrew Elvey-Price, and*the*anonymous reviewer for helpful feedback*on*this work. ...
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