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Confined subgroups of Thompson's group F and its embeddings into wobbling groups
[article]
2018
arXiv
pre-print
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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
2009
Algebraic and Geometric Topology
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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]
2005
arXiv
pre-print
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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]
2009
arXiv
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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]
2019
arXiv
pre-print
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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]
2021
arXiv
pre-print
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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
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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
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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]
2022
arXiv
pre-print
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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]
2003
arXiv
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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
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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
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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]
2014
arXiv
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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
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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
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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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