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

Maksym Chaudkhari
2018 arXiv   pre-print
We obtain a characterisation 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  ... 
arXiv:1809.05146v2 fatcat:e74y7ajncnbkrf6rglyyxahsde

Thompson's groupFand uniformly finite homology

Daniel Staley
2009 Algebraic and Geometric Topology  
This paper seeks to apply 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  ... 
doi:10.2140/agt.2009.9.2349 fatcat:bhan43s3vbev5hmjyttcmtn5wi

Traveller Salesman Property and Richard Thompson's Group F [article]

Victor Guba
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 .  ... 
arXiv:math/0508014v1 fatcat:i45o7xvxnzerhn3k3ymudxdsfu

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

Dan Staley
2009 arXiv   pre-print
This paper demonstrates 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.  ... 
arXiv:0806.2877v2 fatcat:rqrzwu5tjzgxfi7v4fymxlp2yq

On the density of Cayley graphs of R.Thompson's group F in symmetric generators [article]

Victor Guba
2019 arXiv   pre-print
It is known that a 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.  ... 
arXiv:1909.01882v1 fatcat:nvkdodqfovhpxijtpo5gfaufuq

Evacuation schemes on Cayley graphs and non-amenability of groups [article]

Victor Guba
2021 arXiv   pre-print
In this paper we introduce a concept 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.  ... 
arXiv:2112.09812v1 fatcat:vfg26sennfcitbu5mr6qdjv4t4

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]

Dmytro Savchuk
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] .  ... 
arXiv:0803.0043v2 fatcat:mk62j4hocjhfhidt2dnyrgbvji

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

Victor Guba
2022 arXiv   pre-print
We improve some results by Donnelly showing that there exist finite sets Y⊂ 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 .  ... 
arXiv:2201.02308v1 fatcat:7dpjfjwtprfcrpyz6auktl2fli

Thompson's Group F is not Minimally Almost Convex [article]

James Belk, Kai-Uwe Bux
2003 arXiv   pre-print
We prove that 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 .  ... 
arXiv:math/0301141v1 fatcat:5vz7olwwhzfnlkfjxasysratpe

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]

Murray Elder, Cameron Rogers
2016 arXiv   pre-print
We identify two potential sources 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.  ... 
arXiv:1608.06703v2 fatcat:utsebdbtvretnhr6enpnpjljlu

Schreier graphs of actions of Thompson's group F on the unit interval and on the Cantor set [article]

Dmytro Savchuk
2014 arXiv   pre-print
Schreier 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.  ... 
arXiv:1105.4017v3 fatcat:jdtzyltnvzfq5fbdsimqk3bzai

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

Murray Elder, Cameron Rogers
2017 Experimental Mathematics  
In Section 6 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.  ... 
doi:10.1080/10586458.2017.1353452 fatcat:qklu76z55bgdxl7i5ivmtkamci
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