Spectral rigidity of automorphic orbits in free groups

Carette, Mathieu; Francaviglia, Stefano; Kapovich, Ilya; Martino, Armando

doi:10.2140/agt.2012.12.1457

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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

DOI: 10.2140/agt.2012.12.1457

Abstract

It is well-known that a point $T \in {cv}_{N}$ in the (unprojectivized) Culler–Vogtmann Outer space ${cv}_{N}$ is uniquely determined by its translation length function $∥ \cdot ∥_{T} : F_{N} \to ℝ$ . A subset $S$ of a free group $F_{N}$ is called spectrally rigid if, whenever $T, T^{'} \in {cv}_{N}$ are such that $∥ g ∥_{T} = ∥ g ∥_{T^{'}}$ for every $g \in S$ then $T = T^{'}$ in ${cv}_{N}$ . By contrast to the similar questions for the Teichmüller space, it is known that for $N \geq 2$ there does not exist a finite spectrally rigid subset of $F_{N}$ .

In this paper we prove that for $N \geq 3$ if $H \leq Aut (F_{N})$ is a subgroup that projects to a nontrivial normal subgroup in $Out (F_{N})$ then the $H$ –orbit of an arbitrary nontrivial element $g \in F_{N}$ is spectrally rigid. We also establish a similar statement for $F_{2} = F (a, b)$ , provided that $g \in F_{2}$ is not conjugate to a power of $[a, b]$ .

Keywords

marked length spectrum rigidity, free groups, Outer space

Mathematical Subject Classification 2010

Primary: 20E08, 20F65

Secondary: 57M07, 57M50, 53C24

References

Publication

Received: 3 June 2011

Revised: 19 April 2012

Accepted: 2 May 2012

Published: 10 July 2012

Authors

Mathieu Carette
SST/IRMP Chemin du Cyclotron 2 bte L7.01.01 1348 Louvain-la-Neuve Belgium

Stefano Francaviglia
Dipartimento di Matematica of the University of Bologna Piazza di Porta S. Donato 5 40126 Bologna Italy

Ilya Kapovich
Department of Mathematics University of Illinois at Urbana-Champaign 1409 West Green Street Urbana IL 61801 USA

Armando Martino
School of Mathematics University of Southampton Highfield Southampton SO17 1BJ United Kingdom

Volume 12, issue 3 (2012)