Special Issue: International Conference on Semigroups and Groups in Honor of the 65th Birthday of Prof. John RhodesNo Access

ON THE PROPERTIES OF THE CAYLEY GRAPH OF RICHARD THOMPSON'S GROUP F

V. S. GUBA

V. S. GUBA

Vologda State Pedagogical University, 6 S. Orlov Street, Vologda, Russia 160600, Russia

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https://doi.org/10.1142/S021819670400192XCited by:16 (Source: Crossref)

Abstract

We study some properties of the Cayley graph of R. Thompson's group F in generators x₀, x₁. We show that the density of this graph, that is, the least upper bound of the average vertex degree of its finite subgraphs is at least 3. It is known that a 2-generated group is not amenable if and only if the density of the corresponding Cayley graph is strictly less than 4. It is well known this is also equivalent to the existence of a doubling function on the Cayley graph. This means there exists a mapping from the set of vertices into itself such that for some constant K>0, each vertex moves by a distance at most K and each vertex has at least two preimages. We show that the density of the Cayley graph of a 2-generated group does not exceed 3 if and only if the group satisfies the above condition with K=1. Besides, we give a very easy formula to find the length (norm) of a given element of F in generators x₀, x₁. This simplifies the algorithm by Fordham. The length formula may be useful for finding the general growth function of F in generators x₀, x₁ and the growth rate of this function. In this paper, we show that the growth rate of F has a lower bound of .

Keywords:

AMSC: 20F32, 05C25

References

S. I. Adian, Math. USSR Izvestiya 21(3), 425 (1983). Crossref, Google Scholar
M. G. Brin and C. C. Squier, Invent. Math. 79, 485 (1985). Crossref, Web of Science, Google Scholar
K. S. Brown, J. Pure Appl. Algebra 44, 45 (1987). Crossref, Web of Science, Google Scholar
K. S. Brown and R. Geoghegan, Invent. Math. 77, 367 (1984). Crossref, Web of Science, Google Scholar
J. Burillo, Growth of positive words in Thompson's group F, preprint . Google Scholar
J. W. Cannon, W. J. Floyd and W. R. Parry, L'Enseignement Mathématique (2) 42, 215 (1996). Google Scholar
T. Ceccherini-Silberstein, R. Grigorchuk and P. de la Harpe, Proc. Steklov Inst. Math. 1(224), 57 (1999). Google Scholar
S. Cleary and J. Taback, Thompson's group F is not almost convex, preprint . Google Scholar
M. Day, Illinois J. Math. 1, 509 (1957). Crossref, Google Scholar
W. A. Deuber, M. Simonovitz and V. T. Sós, Studia Math. 30, 17 (1995). Web of Science, Google Scholar
E. Følner, Math. Scand. 3, 243 (1955). Crossref, Google Scholar
S. Blake Fordham, Minimal length elements of Thompson's group F, PhD thesis, Brigham Young University (1995) . Google Scholar
F. P. Greenleaf , Invariant Means on Topological Groups and Their Applications ( Van Nostrand , 1969 ) . Google Scholar
R. I. Grigorchuk, Funktsional. Anal. i Prilozhen. 14, 53 (1980). Google Scholar
R. I. Grigorchuk, Sb. Math. 189(1–2), 75 (1998). Crossref, Web of Science, Google Scholar
V. S. Guba, Algorithmic Problems in Groups and Semigroups, eds. J.-C. Birgetet al. (Birkhäuser, Boston–Basel–Berlin, 2000) pp. 91–120. Crossref, Google Scholar
V. S. Guba and M. V. Sapir, J. Austral. Math. Soc. (Ser. A) 62, 315 (1997). Crossref, Google Scholar
V. S. Guba and M. V. Sapir, Memoirs of the Amer. Math. Soc. 130(N 620), 1 (1997). Google Scholar
V. S. Guba and M. V. Sapir, Matem. Sb. 190(8), 3 (1999). Crossref, Google Scholar
H. Kesten, Trans. Amer. Math. Soc. 92, 336 (1959). Crossref, Web of Science, Google Scholar
J. von Neumann, Fund. Math. 13, 73 (1929). Crossref, Google Scholar
A. Yu. Ol'shanskii, Russian Math. Surveys 35(4), 180 (1980). Crossref, Web of Science, Google Scholar
A. Yu. Ol'shanskii and M. V. Sapir, Non-amenable finitely presented torsion-by-cyclic groups, submitted . Google Scholar