Charney's CAT(0) conjecture for Artin groups

Let an Artin group be the group associated to a symmetric matrix (mij)(m_{ij}) with entries in {2,3,}{}\{2,3,\dots\}\cup\{\infty\}, generated by s1,,sns_1,\ldots,s_n subject to the braid relations

sisjsimij factors=sjsisjmij factors.\underbrace{s_is_js_i\cdots}_{m_{ij}\text{ factors}}=\underbrace{s_js_is_j\cdots}_{m_{ij}\text{ factors}}.

Here mij=m_{ij}=\infty means that there is no defining relation between sis_i and sjs_j. Charney's conjecture. Every Artin group is CAT(0)\operatorname{CAT}(0), meaning that it acts properly and cocompactly on a CAT(0)\operatorname{CAT}(0) space. The conjecture has been verified for right-angled Artin groups, namely Artin groups for which every mijm_{ij} equals 22 or \infty; the general assertion is therefore no longer open only in that special case.

Sources & referencesView supporting material

Primary source

Barbara Baumeister, Kai-Uwe Bux, Friedrich Götze, Dawid Kielak and Henning Krause, “Non-crossing partitions”, arXiv:1903.01146 (2019).

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