# Groups, Geometry, and Dynamics

Full-Text PDF (103 KB) | Table of Contents | GGD summary

**Volume 1, Issue 2, 2007, pp. 101–109**

**DOI: 10.4171/GGD/6**

Denominator bounds in Thompson-like groups and flows

Danny Calegari^{[1]}(1) Department of Mathematics, California Institute of Technology, CA 91125, PASADENA, UNITED STATES

Let `T` denote Thompson's group of piecewise 2-adic linear
homeomorphisms of
the circle. Ghys and Sergiescu showed that the
rotation number of every element of `T` is rational, but
their proof is
very indirect. We give here a short, direct proof using train tracks,
which generalizes to
elements of PL^{+}(`S`^{1}) with rational
break points and derivatives
which are powers of some fixed integer, and also to certain
flows on surfaces which we call Thompson-like.
We also obtain an explicit
upper bound on the smallest period of a fixed point in terms of data
which can be read off from the combinatorics of the homeomorphism.

*Keywords: *Thompson's group, rotation number, rationality

Calegari Danny: Denominator bounds in Thompson-like groups and flows. *Groups Geom. Dyn.* 1 (2007), 101-109. doi: 10.4171/GGD/6