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The 6-strand braid group is \(\mathrm{CAT}(0)\). (English) Zbl 1347.20044
Summary: We show that braid groups with at most 6 strands are \(\mathrm{CAT}(0)\) using the close connection between these groups, the associated non-crossing partition complexes, and the embeddability of their diagonal links into spherical buildings of type \(A\). Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is \(\mathrm{CAT}(0)\), giving a partial answer to a conjecture of Brady and McCammond.

MSC:
20F65 Geometric group theory
20F36 Braid groups; Artin groups
20E42 Groups with a \(BN\)-pair; buildings
57M07 Topological methods in group theory
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