×

The topology of local commensurability graphs. (English) Zbl 1499.20077

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.

MSC:

20E26 Residual properties and generalizations; residually finite groups
20E15 Chains and lattices of subgroups, subnormal subgroups
20B99 Permutation groups
20F18 Nilpotent groups
PDFBibTeX XMLCite
Full Text: arXiv Link