The complex volume of SL(n,C)-representations of 3-manifolds

Abstract

Abstract. For a compact 3-manifold M with arbitrary (possibly empty) boundary, we give a parametrization of the set of conjugacy classes of boundary-unipotent representations of π1(M ) into SL(n, C). Our parametrization uses Ptolemy coordinates, which are inspired by coordinates on higher Teichmüller spaces due to Fock and Goncharov. We show that a boundary-unipotent representation determines an element in Neumann's extended Bloch group B(C), and use this to obtain an efficient formula for the Cheeger-Chern-Simons invariant, and in particular for the volume. Computations for the census manifolds show that boundary-unipotent representations are abundant, and numerical comparisons with census volumes, suggest that the volume of a representation is an integral linear combination of volumes of hyperbolic 3-manifolds. This is in agreement with a conjecture of Walter Neumann, stating that the Bloch group is generated by hyperbolic manifolds.

Triangulation independent Ptolemy varieties

“…A convenient language for studying both the gluing equation variety and the Ptolemy variety is that of decorated representations (see [12,9]). We give a brief overview assuming for simplicity that M has a single torus boundary component (the general case and precise definitions are recalled in Section 2.2).…”

“…• A Ptolemy variety P σ (T ), with σ ∈ H 2 (M, ∂M ; Z/2Z), for boundary-unipotent PSL(2, C)-representations with obstruction class to lifting to a boundary-unipotent SL(2, C)-representation given by σ [9,8].…”

“…It is the quotient of P (T ) by a (C * ) c -action, where c is the number of boundary components of M [9,8].…”

“…There is an explicit regular map from each (triangulation dependent) Ptolemy variety to the gluing equation variety, but we shall not need this here (see e.g. [9,7,13]). …”

“…Our goal is to define a refined Ptolemy variety P (T ), which is a topological invariant of M and is guaranteed to detect all irreducible boundary-unipotent SL(2, C)-representations. We also define refined variants of the Ptolemy variety for PSL(2, C)-representations [9,8], and for the enhanced Ptolemy variety [13] for SL(2, C)-representations that are not necessarily boundary-unipotent. The refined variant of the latter detects all irreducible SL(2, C)-representations.…”