Boundaries of Groups and Kleinian Groups — Problem 102

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For $n\ge2$, the $n$-strand braid group has the Artin presentation \[B_n=\left\langle \sigma_1,\ldots,\sigma_{n-1}\ \middle|\ \sigma_i\sigma_j=\sigma_j\sigma_i\ (|i-j|\ge2),\quad \sigma_i\sigma_{i+1}\sigma_i=\sigma_{i+1}\sigma_i\sigma_{i+1}\ (1\le i\le n-2)\right\rangle.\] Set $B_1=\{1\}$. A geodesic metric space $X$ is CAT(0) if, for every geodesic triangle and every two points on its sides, their distance is at most the distance between the corresponding points in the Euclidean triangle with the same side lengths. A discrete group is a CAT(0) group if it acts properly discontinuously and cocompactly by isometries on such a space: for each compact $K\subset X$ only finitely many group elements have $gK\cap K\ne\varnothing$, and the quotient space is compact.

Statement recorded in the supplied source.

Is $B_n$ a CAT(0) group for every $n\ge1$? [S2]

Short English statement

Does every braid group admit a properly discontinuous, cocompact action by isometries on a CAT(0) space?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 6200102 (AMR-061-0102). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Thomas Haettel, Dawid Kielak and Petra Schwer, The 6-Strand Braid Group Is CAT(0), arXiv:1304.5990v2 (2016), Introduction, printed p. 1, and Section 2.4, Definitions 2.27 and 2.29, p. 9. https://arxiv.org/abs/1304.5990v2.