arXiv subjects

Cameron Rogers

Publications and source records attributed to Cameron Rogers.

2 recordsLinked to original sources

arXiv2017

On a theorem of Avez

For each symmetric, aperiodic probability measure $μ$ on a finitely generated group $G$, we define a subset $A_μ$ consisting of group elements $g$ for which the limit of the ratio ${μ^{\ast n}(g)}/{μ^{\ast n}(e)}$ tends to $1$. We prove that $A_μ$ is a subgroup, is amenable, contains every finite normal subgroup, and $G=A_μ$ if and only if $G$ is amenable. For non-amenable groups we show that $A_μ$ is not always a normal subgroup, and can depend on the measure. We formulate some conjectures relating $A_μ$ to the amenable radical.

math.GR↗

arXiv2016

Sub-dominant cogrowth behaviour and the viability of deciding amenability numerically

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 estimating its asymptotic cogrowth rate. 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.

math.GR↗