A virtual geometric action of a braid group (Q6572408)

Let \(B_{4}=\big \langle a,b,c \; \big | \;aba = bab, bcb = cbc, ac = ca \big \rangle\) be the braid on four strings and let \(Z=Z(B_{4}) \simeq \mathbb{Z}\) be the center of \(B_{4}\). The quotient group \(B_{4}/Z\) acts geometrically on a \(\mathrm{CAT}(0)\) space \(X_{0}\) of dimension 2 (see [\textit{T. Brady}, Arch. Math. 63, No. 2, 97--102 (1994; Zbl 0811.20037); Mich. Math. J. 47, No. 2, 313--324 (2000; Zbl 0996.20022)]), this space is a simplicial complex known as the Brady complex, and the corresponding action of \(B_{4}/Z\) is called the standard action.\N\NA first main result in the paper under review is Theorem 1.1: There exists a \(\mathrm{CAT}(0)\) complex \(X_{1}\) of dimension \(2\) with the following two properties (a) \(B_{4}/Z\) acts virtually geometrically on \(X_{1}\); (b) \(B_{4}/Z\) does not admit a properly discontinuous action on \(X_{1}\) by semisimple isometries.\N\NThus, the author constructs a subgroup \(G_{1}\) of finite index in \(B_{4}/Z\) and proves Theorem 1.2: There does not exist a \(G_{1}\)-equivariant map \(f : X_{0} \rightarrow X_{1}\) which is locally injective and locally isometric (up to a constant scaling of the metrics) on the complement of the \(0\)-skeleton of \(X_{0}\). (In this statement, \(G_{1}\) acts geometrically on \(X_{0}\) through the standard action of \(B_{4}/Z\)).\N\NFinally, the author proves in Theorem 1.5 that there exists a \(\mathrm{CAT}(0)\) complex \(Y_{1}\) of dimension \(3\) with the following two properties: (a) \(B_{4}\) acts virtually geometrically on \(Y_{1}\); (b) \(B_{4}\) does not admit a minimal properly discontinuous action on \(Y_{1}\) by semisimple isometries.