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On deformation spaces of nonuniform hyperbolic lattices

Published online by Cambridge University Press:  03 May 2016

SUNGWOON KIM
Affiliation:
Department of Mathematics, Jeju National University, Jeju, Republic of Korea. e-mail: sungwoon@jejunu.ac.kr
INKANG KIM
Affiliation:
School of Mathematics, Korea Institute for Advanced Study, Seoul, Republic of Korea. e-mail: inkang@kias.re.kr
Corresponding

Abstract

Let Γ be a nonuniform lattice acting on the real hyperbolic n-space. We show that in dimension greater than or equal to 4, the volume of a representation is constant on each connected component of the representation variety of Γ in SO(n, 1). Furthermore, in dimensions 2 and 3, there is a semialgebraic subset of the representation variety such that the volume of a representation is constant on connected components of the semialgebraic subset. Combining our approach with the main result of [2] gives a new proof of the local rigidity theorem for nonuniform hyperbolic lattices and the analogue of Soma's theorem, which shows that the number of orientable hyperbolic manifolds dominated by a closed, connected, orientable 3-manifold is finite, for noncompact 3-manifolds.

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Research Article
Copyright
Copyright © Cambridge Philosophical Society 2016 

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