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Variations on a theme of Horowitz

Variations on a theme of Horowitz
Variations on a theme of Horowitz
Horowitz showed that for every n at least 2, there exist elements w1, ..., wn in the free group F =free(a,b) which generate non-conjugate maximal cyclic subgroups of F and which have the property that trace(f(w1)) = ... =trace(f(wn)) for all faithful representations f of F into SL(2, C). Randol used this result to show that the length spectrum of a hyperbolic surface has unbounded multiplicity. Masters has recently extended this unboundness of the length spectrum to hyperbolic 3-manifolds. The purpose of this note is to present a survey of what is known about characters of faithful representations of F into SL(2, C), to give a conjectural topological characterization of such n-tuples of elements of F, and to discuss the case of faithful representations of general surface groups and 3-manifold groups.
0521540135
307-341
Anderson, J.W.
739c0e33-ef61-4502-a675-575d08ee1a98
Anderson, J.W.
739c0e33-ef61-4502-a675-575d08ee1a98

Anderson, J.W. (2003) Variations on a theme of Horowitz. London Mathematical Society Lecture Note Series, 299, 307-341.

Record type: Article

Abstract

Horowitz showed that for every n at least 2, there exist elements w1, ..., wn in the free group F =free(a,b) which generate non-conjugate maximal cyclic subgroups of F and which have the property that trace(f(w1)) = ... =trace(f(wn)) for all faithful representations f of F into SL(2, C). Randol used this result to show that the length spectrum of a hyperbolic surface has unbounded multiplicity. Masters has recently extended this unboundness of the length spectrum to hyperbolic 3-manifolds. The purpose of this note is to present a survey of what is known about characters of faithful representations of F into SL(2, C), to give a conjectural topological characterization of such n-tuples of elements of F, and to discuss the case of faithful representations of general surface groups and 3-manifold groups.

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Published date: 2003

Identifiers

Local EPrints ID: 29876
URI: https://eprints.soton.ac.uk/id/eprint/29876
ISBN: 0521540135
PURE UUID: b2f5d990-3753-4dfd-8f4b-cefaf48fde07
ORCID for J.W. Anderson: ORCID iD orcid.org/0000-0002-7849-144X

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Date deposited: 12 May 2006
Last modified: 06 Jun 2018 13:03

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