Groups, Geometry, and Dynamics
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Denominator bounds in Thompson-like groups and flowsDanny Calegari (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+(S1) 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