*Soficity, recurrence and short cycles of exponential
maps (joint work with K. Juschenko)
*

Abstract

Let $f$ be an exponentiation map mod $p$, or, more precisely, the map
from $\{0,1,...p-1\}$ to itself defined by $f(x) \equiv 2^x \mod p$.
It is easy to show that $f$ and $f\circ f$ have few fixed points.
Showing that $f\circ f\circ f$ has $o(p)$ fixed points is harder, and
was open; we show how to prove it. What about $f\circ f\circ f\circ f$?
There, the problem is still open; we show its connection to
{\em sofic groups}. More precisely: if the Higman group is sofic,
then there is a map $f$ that (a) behaves almost everywhere like an
exponentially map, and (b) satisfies $f(f(f(f(x))))=x$ for almost all x.
The proof rests in part on an elementary proof of a special case of the
uniqueness of sofic representations of amenable groups.