Integrality of volumes of representations

Michelle Bucher,Marc Burger,Alessandra Iozzi

doi:10.1007/s00208-020-02047-9

Michelle Bucher, Marc Burger + Show 1 more

Open Access

https://doi.org/10.1007/s00208-020-02047-9

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Journal: Mathematische Annalen	Publication Date: Feb 24, 2021
Citations: 1	License type: open-access

Affiliation: University of Geneva, ETH Zurich

Abstract

Let M be an oriented complete hyperbolic n-manifold of finite volume. Using the definition of volume of a representation previously given by the authors in [3] we show that the volume of a representation rho :pi _1(M)rightarrow mathrm {Isom}^+({{mathbb {H}}}^n), properly normalized, takes integer values if n is even and ge 4. If M is not compact and 3-dimensional, it is known that the volume is not locally constant. In this case we give explicit examples of representations with volume as arbitrary as the volume of hyperbolic manifolds obtained from M via Dehn fillings.

Highlights

Let M be a connected oriented complete hyperbolic manifold of finite volume, which we represent as the quotient M = \Hn of real hyperbolic n-space Hn by a torsionfree lattice < Isom+(Hn) in the group of orientation preserving isometries of Hn
Given a representation ρ : → Isom+(Hn), our central object of study is the volume Vol(ρ) of ρ as defined in [6] for n = 2 and in general in [3]. This notion extends the classical one introduced in [15] for M compact and, as it was shown in [20], if M is of finite volume it coincides with definitions introduced by other authors [10,12,19]
Using that R → R/Z admits a bounded Borel section, one obtains readily, both for the trivial and the nontrivial modules, long exact sequences in Borel and in bounded Borel cohomology with commutative squares coming from the comparison maps cZ and cR between these two cohomology theories:

Summary

Introduction

Given a representation ρ : → Isom+(Hn), our central object of study is the volume Vol(ρ) of ρ as defined in [6] for n = 2 and in general in [3] This notion extends the classical one introduced in [15] for M compact and, as it was shown in [20], if M is of finite volume it coincides with definitions introduced by other authors [10,12,19]. In the Appendix we prove the continuity of the map ρ → Vol(ρ)

Various cohomology theories

Proportionality between volume and Euler class

The bounded Euler class

Definition of volume and congruence relations

Examples of nontrivial and non-maximal representations

Dimension 3: representations given by Dehn filling

Representations giving rational multiples of the maximal representation