The geometric combinatorics of k-shuffle braid groups
Abstract
This paper is the first in a series of two which aims at introducing a new family of groups called k-shuffle braid groups, denoted ℬn(k). These groups have two intrinsically related stories. They were first observed by Calegari in arXiv:2105.11265 by looking at spaces of complex monic polynomials of degree n whose critical values lie in ℂ ∖ {1, e2πi/k, …, e(2π(k−1))/k}. The k-shuffle braid groups are the fundamental groups of those spaces, which correspond geometrically to complements of k translates of the discriminant variety. The second story starts from the poset of k-shuffle partitions, which were introduced by Drew Armstrong and explored in depth in his thesis arXiv:math/0611106. In this paper, we focus on the combinatorial aspects of k-shuffle braid groups starting from the poset of k-shuffle non-crossing partitions. We build a CW-complex on which the groups act freely, properly discontinuously and cocompactly, which we show is CAT(0) for dimension up to 7. For higher dimension, we show that the CAT(0) property is implied by the conjecture of Brady and McCammond that the dual braid complex is CAT(0) (arXiv:0909.4778).
Related
- The geometric combinatorics of k-shuffle polynomials
- Braided Species, Hopf Monoids, and the Combinatorics of the Braid Arrangement
- A virtual geometric action of a braid group
- Disconnectedness of stability manifolds and their geometric loci, and separation of braid orbits of semiorthogonal decompositions
- On unifying Jones representations of braid groups and Thompson's group
BibTeX
@misc{postelvinay2026geometric,
title = {{The geometric combinatorics of $k$-shuffle braid groups}},
author = {Chloé Postel-Vinay},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.28986},
eprint = {2610.28986},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.28986},
}
RIS
TY - UNPB
TI - The geometric combinatorics of k-shuffle braid groups
AU - Chloé Postel-Vinay
PY - 2026
DA - 2026/10/20
PB - arXiv
DO - 10.48550/arxiv.2610.28986
UR - https://arxiv.org/abs/2610.28986
AB - This paper is the first in a series of two which aims at introducing a new family of groups called k-shuffle braid groups, denoted ℬ_n(k). These groups have two intrinsically related stories. They were first observed by Calegari in arXiv:2105.11265 by looking at spaces of complex monic polynomials of degree n whose critical values lie in ℂ ∖ {1, e^2πi/k, …, e^(2π(k−1))/k}. The k-shuffle braid groups are the fundamental groups of those spaces, which correspond geometrically to complements of k translates of the discriminant variety. The second story starts from the poset of k-shuffle partitions, which were introduced by Drew Armstrong and explored in depth in his thesis arXiv:math/0611106. In this paper, we focus on the combinatorial aspects of k-shuffle braid groups starting from the poset of k-shuffle non-crossing partitions. We build a CW-complex on which the groups act freely, properly discontinuously and cocompactly, which we show is CAT(0) for dimension up to 7. For higher dimension, we show that the CAT(0) property is implied by the conjecture of Brady and McCammond that the dual braid complex is CAT(0) (arXiv:0909.4778).
ER -