Counting growth types of automorphisms of free groups

2008/01/31 by Gilbert Levitt · 1 citation
Mathematics · #math.GR #math.GT #msc:20E05 #msc:20F65

paper · pdf

published as Geom. Funct. Anal. 19 (2009), no. 4, 1119--1146 · final version, to appear in GAFA; proof of 3.1 simplified thanks to the referee

arxiv created 2008/10/06 · arxiv updated 2019/06/07

Abstract

Given an automorphism of a free group F_n, we consider the following invariants: e is the number of exponential strata (an upper bound for the number of different exponential growth rates of conjugacy classes); d is the maximal degree of polynomial growth of conjugacy classes; R is the rank of the fixed subgroup. We determine precisely which triples (e,d,R) may be realized by an automorphism of F_n. In particular, the inequality e≤ (3n-2)/4 (due to Levitt-Lustig) always holds. In an appendix, we show that any conjugacy class grows like a polynomial times an exponential under iteration of the automorphism.

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