Bou-Rabee, Khalid; Studenmund, Daniel The topology of local commensurability graphs. (English) Zbl 1499.20077 New York J. Math. 24, 429-442 (2018). Summary: We initiate the study of the \(p\)-local commensurability graph of a group, where \(p\) is a prime. This graph has vertices consisting of all finite-index subgroups of a group, where an edge is drawn between \(A\) and \(B\) if [\(A: A \cap B]\) and \([B: A \cap B]\) are both powers of \(p\). We show that any component of the \(p\)-local commensurability graph of a group with all nilpotent finite quotients is complete. Further, this topological criterion characterizes such groups. In contrast to this result, we show that for any prime \(p\) the \(p\)-local commensurability graph of any large group (e.g. a nonabelian free group or a surface group of genus two or more or, more generally, any virtually special group) has geodesics of arbitrarily long length. Cited in 1 Document MSC: 20E26 Residual properties and generalizations; residually finite groups 20E15 Chains and lattices of subgroups, subnormal subgroups 20B99 Permutation groups 20F18 Nilpotent groups Keywords:commensurability; nilpotent groups; free groups; very large groups PDFBibTeX XMLCite \textit{K. Bou-Rabee} and \textit{D. Studenmund}, New York J. Math. 24, 429--442 (2018; Zbl 1499.20077) Full Text: arXiv Link