Simple length rigidity for Kleinian surface groups and applications

Abstract

We prove that a Kleinian surface groups is determined, up to conjugacy in the isometry group of 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, hyperblizable 3-manifold M is similarly determined by the translation lengths of images of elements of π1(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.

Finite step Rigidity of Hitchin representations and special Margulis-Smilga spacetimes

“…Moreover, in a recent work [BCL20] Bridgeman-Canary-Labourie show that two Hitchin representations are conjugate to each other if and only if they have the same marked trace spectra over simple non-separating elements. Similar results involving infinite data but only over simple non-separating elements were also obtained by Brideman-Canary [BC17] in the context of representations of surface groups inside PSL 2 (C) and by Duchin-Leininger-Rafi [DLR10] in the context of flat surfaces.…”

Simple length rigidity for Hitchin representations

“…Our proof follows the outline suggested by the proof in Bridgeman-Canary [5] that the isometry group of the intersection function on quasifuchsian space is generated by the extended mapping class group and complex conjugation.…”

“…Bridgeman, Canary, Labourie and Sambarino [6] proved that Hitchin representations, are determined up to conjugacy in PGL d (R) by the spectral radii of all elements. Bridgeman and Canary [5] proved that discrete faithful representations of π 1 (S) into PSL(2, C) are determined by the translation lengths of simple non-separating curves on S. Duchin, Leininger and Rafi [12] showed that the simple marked length spectrum determines a flat surface, but that no collection of finitely many simple closed curves suffices to determine a flat surface. On the other hand, Marché and Wolff [26,Section 3] gave examples of non-conjugate, indiscrete, non-elementary representations of a closed surface group of genus two into PSL 2 (R) with the same simple marked length spectra.…”

An introduction to pressure metrics for higher Teichmüller spaces

“…Bridgeman and Canary [15] have shown that the group of diffeomorphisms of quasifuchsian space QF (S) which preserve the renormalized intersection number is generated by the extended mapping class group and complex conjugation. So one may also consider the corresponding analogue of Problem 2 in quasifuchsian space.…”