Random groups have fixed points on \(\text{CAT}(0)\) cube complexes
The random groups considered in this paper are built using Gromov's graph model. Specifically, let \(G=(V,E)\) be a finite oriented graph. To each edge one associates a random element of the generating set \(\{s_1,\dots,s_k, s_1^{-1},\dots, s_k^{-1}\}\); these choices are made independently and with uniform probability distribution. Each unoriented cycle in \(G\) gives rise to a relation between the generators in a natural way. Thus, one obtains \(\Gamma(G)\), the \(k\)-generated random group associated to \(G\).NEWLINENEWLINEOf particular interest in this context are the graphs that form a sequence of expanders \(\{G_n\}\). By definition, such graphs have uniformly bounded vertex degree, satisfy a uniform quadratic Poincaré inequality, and the number of their vertices tends to infinity. Under additional conditions on the girth and degree of \(G_n\), the corresponding random groups have Kazhdan's property (T) with high probability, see [\textit{M. Gromov}, Geom. Funct. Anal. 13, No. 1, 73--146 (2003; Zbl 1122.20021)] and [\textit{L. Silberman}, Geom. Funct. Anal. 13, No. 1, 147--177 (2003; Zbl 1124.20027)]. Recall that the property (T) is equivalent to the existence of a common fixed point for every isometric action of \(\Gamma\) on a Hilbert space.NEWLINENEWLINEThe main result of this paper (Theorem 1.5) replaces the Hilbert space by a complete CAT(0) cube complex. The action is not assumed to be simplicial, i.e., it need not preserve individual simplices. Theorem 1.5 states that for any sequence of expanders of vertex degree at least \(2\) and sufficiently large girth, every isometric action of the random group on a complete CAT(0) cube complex has a common fixed point, with high probability.
scientific article; zbMATH DE number 6024109
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