Fast growth in the Følner function for Thompson's group \(F\). (Q374104)

Thompson's group \(F\) is a much-studied infinite group which can be defined in several ways, the most common being as a particular group of homeomorphisms of the unit interval. It has many interesting properties and in recent years the question whether or not \(F\) is amenable was widely studied.NEWLINENEWLINE The paper under review contributes to this question by studying Følner functions for Thompson's group \(F\). Given a group \(G\) generated by a finite set \(\Gamma\) a finite subset \(A\) of \(G\) is called \(\varepsilon\)-Følner if NEWLINE\[NEWLINE\sum_{\gamma\in\Gamma}|A\cdot\gamma\bigtriangleup A|<\varepsilon|A|,NEWLINE\]NEWLINE where \(\bigtriangleup\) denotes the symmetric difference. Using this, one defines the Følner function of \(G\) with respect to \(\Gamma\) by NEWLINE\[NEWLINE\mathrm{F{\o}l}_{G,\Gamma}(n)=\min\{|A|:A\subseteq G\text{ is }\tfrac{1}{n}\text{-Følner with respect to }\Gamma\}NEWLINE\]NEWLINE and \(\mathrm{F{\o}l}_{G,\Gamma}(n)=\infty\) if there is no \(1/n\)-Følner set with respect to \(\Gamma\). So, according to the Følner criterion, \(G\) is amenable if and only if its Følner function with respect to some finite generating set (hence with respect to all finite generating sets) is finite valued.NEWLINENEWLINE The author derives the following lower bound for the Følner function of Thompson's group \(F\): For any finite generating set \(\Gamma\) of \(F\) there exists a constant \(C>1\) such that \(\mathrm{F{\o}l}_{F,\Gamma}(C^n)\geq\exp_n(0)\). Here, \(\exp_p(k)\) denotes the \(p\)-fold composition of the exponential function defined by \(\exp_0(k)=k\) and \(\exp_{p+1}(k)=2^{\exp_p(k)}\).NEWLINENEWLINE As main tool for this lower bound the notion of marginal sets is introduced. Roughly speaking, marginal sets are sets that hardly intersect Følner sets. The author studies the partial right action of \(F\) on the set of finite rooted ordered binary trees and using this action he identifies marginal sets in \(F\).