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Marked length rigidity for Fuchsian buildings

Published online by Cambridge University Press:  13 March 2018

DAVID CONSTANTINE  and
JEAN-FRANÇOIS LAFONT
DAVID CONSTANTINE
Affiliation:
Wesleyan University, Mathematics and Computer Science Department, Middletown, CT 06459, USA email dconstantine@wesleyan.edu
JEAN-FRANÇOIS LAFONT
Affiliation:
Department of Mathematics, Ohio State University, Columbus, OH 43210, USA email jlafont@math.ohio-state.edu
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Abstract

We consider finite $2$-complexes $X$ that arise as quotients of Fuchsian buildings by subgroups of the combinatorial automorphism group, which we assume act freely and cocompactly. We show that locally CAT($-1$) metrics on $X$, which are piecewise hyperbolic and satisfy a natural non-singularity condition at vertices, are marked length spectrum rigid within certain classes of negatively curved, piecewise Riemannian metrics on $X$. As a key step in our proof, we show that the marked length spectrum function for such metrics determines the volume of $X$.

Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 39 , Issue 12 , December 2019 , pp. 3262 - 3291
Copyright
© Cambridge University Press, 2018 

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