$C^{*}$ Estimates for Averaging Sums of Elements in the Thompson Group $F$

Details $C^{*}$ Estimates for Averaging Sums of Elements in the Thompson Group $F$ $C^{*}$ Estimates for Averaging Sums of Elements in the Thompson Group $F$

2006-03-07

arxiv.org/abs/math/0603168v1

Mathematics

Group Theory

submitted to the Proceedings of the OAMP3 conference, Bucharest, August, 2005

Scientific paper

In this paper we study the non-amenability question of the Thompson group $F$ from the $C^{*}$ algebra side. Using a characterization of amenability in this framework we set about evaluating the reduced norm of the averages $\frac{1}{n}\sum x_0^ix_1x_0^{-i}$, where $x_0$ and $x_1$ are the generators of $F$ in its finite presentation. We prove that when $n$ is sufficiently large the above norm concentrates on a specific subset of $F$, easy to describe using the new normal form for elements in $F$, found by Guba and Sapir. We view this subset as the only obstruction against non-amenability.

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