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Reflections on branched covers
Abstract
A flag complex L determines a locally CAT(0) cube complex PL with the links of all vertices isomorphic to L. The fundamental group of PL is the commutator subgroup of the right-angled Coxeter group WL. We observe that an edge contraction L → L/e determines a $2$-fold branched cover PL → PL/e (up to a homotopy equivalence). We use this branched cover to give an inequality between the 𝔽2-L2-Betti numbers of WL and WL/e. This reduces an 𝔽2-version of the Singer Conjecture for right-angled Coxeter groups to minimal flag triangulations of even dimensional spheres. We also use this inequality to compute the L2-Betti numbers of WL when L is a flag triangulation of the projective plane or the torus.
Related
- Xiao's Genus-Two Fibration: Branched Covers and Braid Monodromy
- Stable rational cohomology splitting of moduli of branched covers of curves
- Exact triangle and spectral sequence of the equivariant Floer homology of double branched covers
- Pullback metrics and homological obstructions to BLD remetrization of branched covers
- Branched Covers and Fiberwise Degeneration along Weak Kähler Geodesics
BibTeX
@misc{avramidi2026reflections,
title = {{Reflections on branched covers}},
author = {Grigori Avramidi and Boris Okun and Kevin Schreve},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.02723},
eprint = {2610.02723},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.02723},
}
RIS
TY - UNPB
TI - Reflections on branched covers
AU - Grigori Avramidi
AU - Boris Okun
AU - Kevin Schreve
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.02723
UR - https://arxiv.org/abs/2610.02723
AB - A flag complex L determines a locally CAT(0) cube complex P_L with the links of all vertices isomorphic to L. The fundamental group of P_L is the commutator subgroup of the right-angled Coxeter group W_L. We observe that an edge contraction L → L/e determines a $2$-fold branched cover P_L → P_L/e (up to a homotopy equivalence). We use this branched cover to give an inequality between the 𝔽_2-L^2-Betti numbers of W_L and W_L/e. This reduces an 𝔽_2-version of the Singer Conjecture for right-angled Coxeter groups to minimal flag triangulations of even dimensional spheres. We also use this inequality to compute the L^2-Betti numbers of W_L when L is a flag triangulation of the projective plane or the torus.
ER -