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Commentarii Mathematici Helvetici

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Volume 92, Issue 4, 2017, pp. 715–750
DOI: 10.4171/CMH/422

Published online: 2017-10-24

Simple length rigidity for Kleinian surface groups and applications

Martin Bridgeman[1] and Richard D. Canary[2]

(1) Boston College, Chestnut Hill, USA
(2) University of Michigan, Ann Arbor, USA

We prove that a Kleinian surface group is determined, up to conjugacy in the isometry group of $\mathbb H^3$, by its simple marked length spectrum. As a first application, we show that a discrete faithful representation of the fundamental group of a compact, acylindrical, hyperbolizable 3-manifold $M$ is similarly determined by the translation lengths of images of elements of $\pi_1(M)$ represented by simple curves on the boundary of $M$. As a second application, we show the group of diffeomorphisms of quasifuchsian space which preserve the renormalized pressure intersection is generated by the (extended) mapping class group and complex conjugation.

Keywords: Kleinian surface groups, length spectrum and rigidity

Bridgeman Martin, Canary Richard: Simple length rigidity for Kleinian surface groups and applications. Comment. Math. Helv. 92 (2017), 715-750. doi: 10.4171/CMH/422