On the density of Cayley graphs of R.Thompson’s group F in symmetric generators
| Autor: | Victor Guba |
|---|---|
| Rok vydání: | 2021 |
| Předmět: |
Set (abstract data type)
Finite graph Combinatorics Cayley graph Group (mathematics) Computer Science::Information Retrieval General Mathematics Astrophysics::Instrumentation and Methods for Astrophysics Computer Science::General Literature Computer Science::Computation and Language (Computational Linguistics and Natural Language and Speech Processing) Infimum and supremum Mathematics |
| Zdroj: | International Journal of Algebra and Computation. 31:969-981 |
| ISSN: | 1793-6500 0218-1967 |
| Popis: | By the density of a finite graph we mean its average vertex degree. For an [Formula: see text]-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. It is known that a group with [Formula: see text] generators is amenable if and only if the density of the corresponding Cayley graph equals [Formula: see text]. A famous problem on the amenability of R. Thompson’s group [Formula: see text] is still open. Due to the result of Belk and Brown, it is known that the density of its Cayley graph in the standard set of group generators [Formula: see text], is at least [Formula: see text]. This estimate has not been exceeded so far. For the set of symmetric generators [Formula: see text], where [Formula: see text], the same example only gave an estimate of [Formula: see text]. There was a conjecture that for this generating set equality holds. If so, [Formula: see text] would be non-amenable, and the symmetric generating set would have the doubling property. This would mean that for any finite set [Formula: see text], the inequality [Formula: see text] holds. In this paper, we disprove this conjecture showing that the density of the Cayley graph of [Formula: see text] in symmetric generators [Formula: see text] strictly exceeds [Formula: see text]. Moreover, we show that even larger generating set [Formula: see text] does not have doubling property. |
| Databáze: | OpenAIRE |
| Externí odkaz: |
|