Bounded cohomology and optimal separating constant for representations
Abstract
Let Σ be a closed oriented surface of genus >1 and M a complete hyperbolic 3-manifold with a marking i:Σ⟶ M. We consider the case that M has no parabolic cusps and at least one of the two ends is simply degenerate. For varGamma=π1(Σ), let ρM:varGamma⟶ PSL2(ℂ) be the holonomy of M and ρ:varGamma⟶ PSL2(ℂ) any representation in PSL2(ℂ). We will show that, if ρ is discrete and non-faithful, then |[Vol(ρ)]−[Vol(ρM)]|∞≥ v3 holds, where [Vol(ρ)] denotes the bounded fundamental class of ρ in the bounded cohomology Hb3(varGamma,ℝ) of varGamma and v3 is the volume of a regular ideal 3-simplex in ℍ3. As an application, we present a rigidity theorem for ρM in the set of representations ρ of varGamma in PSL2(ℂ) in terms of [Vol(ρ)]. The rigidity theorem implies that v3 is the optimal separating constant.
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BibTeX
@misc{nakano2026bounded,
title = {{Bounded cohomology and optimal separating constant for representations}},
author = {Yushi Nakano and Teruhiko Soma},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.06937},
eprint = {2609.06937},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.06937},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bounded cohomology and optimal separating constant for representations
AU - Yushi Nakano
AU - Teruhiko Soma
PY - 2026
DA - 2026/09/07
PB - arXiv
DO - 10.48550/arxiv.2609.06937
UR - https://arxiv.org/abs/2609.06937
AB - Let Σ be a closed oriented surface of genus >1 and M a complete hyperbolic 3-manifold with a marking i:Σ⟶ M. We consider the case that M has no parabolic cusps and at least one of the two ends is simply degenerate. For varGamma=π_1(Σ), let ρ_M:varGamma⟶ PSL_2(ℂ) be the holonomy of M and ρ:varGamma⟶ PSL_2(ℂ) any representation in PSL_2(ℂ). We will show that, if ρ is discrete and non-faithful, then |[Vol(ρ)]−[Vol(ρ_M)]|_∞≥ v_3 holds, where [Vol(ρ)] denotes the bounded fundamental class of ρ in the bounded cohomology H_b^3(varGamma,ℝ) of varGamma and v_3 is the volume of a regular ideal 3-simplex in ℍ^3. As an application, we present a rigidity theorem for ρ_M in the set of representations ρ of varGamma in PSL_2(ℂ) in terms of [Vol(ρ)]. The rigidity theorem implies that v_3 is the optimal separating constant.
ER -