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Spectral rigidity of automorphic orbits in free groups

Mathieu Carette, Stefano Francaviglia, Ilya Kapovich and Armando Martino

Algebraic & Geometric Topology 12 (2012) 1457–1486

It is well-known that a point T cvN in the (unprojectivized) Culler–Vogtmann Outer space cvN is uniquely determined by its translation length function T: FN . A subset S of a free group FN is called spectrally rigid if, whenever T,T cvN are such that gT = gT for every g S then T = T in cvN. By contrast to the similar questions for the Teichmüller space, it is known that for N 2 there does not exist a finite spectrally rigid subset of FN.

In this paper we prove that for N 3 if H Aut(FN) is a subgroup that projects to a nontrivial normal subgroup in Out(FN) then the H–orbit of an arbitrary nontrivial element g FN is spectrally rigid. We also establish a similar statement for F2 = F(a,b), provided that g F2 is not conjugate to a power of [a,b].

marked length spectrum rigidity, free groups, Outer space
Mathematical Subject Classification 2010
Primary: 20E08, 20F65
Secondary: 57M07, 57M50, 53C24
Received: 3 June 2011
Revised: 19 April 2012
Accepted: 2 May 2012
Published: 10 July 2012
Mathieu Carette
Chemin du Cyclotron 2
bte L7.01.01
1348 Louvain-la-Neuve
Stefano Francaviglia
Dipartimento di Matematica of the University of Bologna
Piazza di Porta S. Donato 5
40126 Bologna
Ilya Kapovich
Department of Mathematics
University of Illinois at Urbana-Champaign
1409 West Green Street
Urbana IL 61801
Armando Martino
School of Mathematics
University of Southampton
SO17 1BJ
United Kingdom