arXiv subjects
Publications and source records attributed to Pere Menal-Ferrer.
Schwarz's Lemma leads to a natural interpolation problem for analytic functions from the disc into itself. The corresponding interpolating sequences are geometrically described in terms of a certain hyperbolic density.
For a complete hyperbolic three manifold M, we consider the representations of its fundamental group obtained by composing a lift of the holonomy with complex finite dimensional representations of SL(2,C). We prove a vanishing result for the cohomology of M with coefficients twisted by these representations, using techniques of Matsushima-Murakami. We give some applications to local rigidity.
For an oriented finite volume hyperbolic 3-manifold M with a fixed spin structure η, we consider a sequence of invariants {τ_n(M; η)}. Roughly speaking, {τ_n(M; η)} is the Reidemeister torsion of M with respect to the representation given by the composition of the lift of the holonomy representation defined by η, and the n-dimensional, irreducible, complex representation of SL(2,C). In the present work, we focus on two aspects of this invariant: its asymptotic behavior and its relationship with the complex-length spectrum of the manifold. Concerning the former, we prove that for suitable spin structures, log(τ_n(M; η)) grows as -n^2 Vol(M)/4π, extending thus the result obtained by W. Mueller for the compact case. Concerning the latter, we prove that the sequence {τ_n(M; η)} determines the complex-length spectrum of the manifold up to complex conjugation.
Given a finite volume hyperbolic 3-manifold, we compose a lift of the holonomy in SL(2,C) with the n-dimensional irreducible representation of SL(2,C) in SL(n,C). In this paper we give local coordinates of the SL(n,C)-character variety around this representation. As a corollary, this representation is isolated among all representations that are unipotent at the cusps.
Given a hyperbolic knot, we prove that the Reidemeister torsion of any lift of the holonomy to SL(2,C) is invariant under mutation along a Conway sphere.