Title: Numerical studies of Thompson's group F and related groups,
Abstract: We have developed polynomial-time algorithms to generate terms of the
cogrowth series for groups $\mathbb{Z}\wr \mathbb{Z},$ the lamplighter group,
$(\mathbb{Z}\wr \mathbb{Z})\wr \mathbb{Z}$ and the Navas-Brin group $B.$ We
have also given an improved algorithm for the coefficients of Thompson's group
$F,$ giving 32 terms of the cogrowth series. We develop numerical techniques to
extract the asymptotics of these various cogrowth series. We present improved
rigorous lower bounds on the growth-rate of the cogrowth series for Thompson's
group $F$ using the method from \cite{HHR15} applied to our extended series. We
also generalise their method by showing that it applies to loops on any locally
finite graph. Unfortunately, lower bounds less than 16 do not help in
determining amenability.
Again for Thompson's group $F$ we prove that, if the group is amenable, there
cannot be a sub-dominant stretched exponential term in the
asymptotics\footnote{ }. Yet the numerical data provides compelling evidence
for the presence of such a term. This observation suggests a potential path to
a proof of non-amenability: If the universality class of the cogrowth sequence
can be determined rigorously, it will likely prove non-amenability.
We estimate the asymptotics of the cogrowth coefficients of $F$ to be $$ c_n
\sim c \cdot \mu^n \cdot \kappa^{n^\sigma \log^\delta{n}} \cdot n^g,$$ where
$\mu \approx 15,$ $\kappa \approx 1/e,$ $\sigma \approx 1/2,$ $\delta \approx
1/2,$ and $g \approx -1.$ The growth constant $\mu$ must be 16 for amenability.
These two approaches, plus a third based on extrapolating lower bounds, support
the conjecture \cite{ERvR15, HHR15} that the group is not amenable.