The authors of this article develop the basic theory of (abstract) commensurators in the context of profinite groups and with applications to the study of totally disconnected locally compact groups.
The commensurator \(\mathrm{Comm}(G)\) of a profinite group \(G\) is defined as follows. A virtual automorphism of \(G\) is a continuous isomorphism between open subgroups of \(G\). Two virtual automorphisms of \(G\) are equivalent if they coincide on some open subgroup of \(G\). The group \(\mathrm{Comm}(G)\) consists of equivalence classes of virtual automorphisms of \(G\), with multiplication given by composition of suitable representatives.
The concept is motivated by the related notion of relative commensurators, groups which arise, for instance, in the study of lattices in algebraic groups over local fields and automorphism groups of trees. The commensurator \(\mathrm{Comm}(G)\) plays a role analogous to the automorphism group of \(G\) when one is interested in properties of the (abstract) commensurability class of \(G\) rather than the specific group \(G\) itself. One can equip \(\mathrm{Comm}(G)\) with two different topologies, the \(\operatorname{Aut}\)-topology and the strong topology, each rendering it into a topological group.
The \(\operatorname{Aut}\)-topology on \(\mathrm{Comm}(G)\) is a generalisation of the standard topology on the automorphism group \(\operatorname{Aut}(G)\), arising from the natural homomorphisms \(\operatorname{Aut}(U)\to\mathrm{Comm}(G)\) for open subgroups \(U\) of \(G\). The \(\operatorname{Aut}\)-topology is useful for understanding the structure of the group \(\mathrm{Comm}(G)\): in many examples, where \(\mathrm{Comm}(G)\) turns out to be isomorphic to a familiar locally compact group, the \(\operatorname{Aut}\)-topology coincides with the ‘natural’ topology. For instance, if \(G\) is a compact \(p\)-adic Lie group, then \(\mathrm{Comm}(G)\), equipped with the \(\operatorname{Aut}\)-topology, is isomorphic to the automorphism group of the \(\mathbb Q_p\)-Lie algebra associated to \(G\). In examples where \(\mathrm{Comm}(G)\) is not locally compact with respect to the \(\operatorname{Aut}\)-topology, e.g.when \(G\) is a non-Abelian free pro-\(p\) group, it appears to be difficult to describe \(\mathrm{Comm}(G)\).
The strong topology on \(\mathrm{Comm}(G)\) is a suitable tool for studying so-called envelopes of \(G\). It is known that a topological group \(L\) is totally disconnected locally compact if and only if it contains an open profinite subgroup. If \(L\) contains the profinite group \(G\) as an open subgroup, one calls \(L\) an envelope of \(G\). The commensurator \(\mathrm{Comm}(G)\) is closely related to the problem of describing possible envelopes \(L\) of \(G\): such an envelope \(L\) can often be recovered from the natural homomorphism \(L\to\mathrm{Comm}(G)\). Of particular interest is the case where \(G\) admits a topologically simple envelope. A fundamental open problem in this context is due to G. A.Willis [J. Algebra 312, No. 1, 405–417 (2007; Zbl 1119.22005)]: is it true that a profinite group \(G\) admits, up to isomorphism, at most one compactly generated topologically simple envelope?
In the paper under review many examples of profinite groups and their commensurators are considered and put into context. These include the Nottingham group, the pro-\(2\) completion of the Grigorchuk group, open compact subgroups of simple algebraic groups over local fields and absolute Galois groups. The commensurator of the Nottingham group has been determined by M. Ershov [in Trans. Am. Math. Soc. 362, No. 12, 6663–6678 (2010; Zbl 1225.20026)]; his result implies that the Nottingham group does not admit any topologically simple envelopes. In contrast, based on work of C. E.Röver [Geom. Dedicata 94, 45–61 (2002; Zbl 1064.20032)] on the commensurator of the Grigorchuk group, one can construct a topologically simple envelope of the pro-\(2\) completion of the Grigorchuk group. Several known important results, such as Pink’s analogue of Mostow’s strong rigidity theorem for simple algebraic groups over local fields and the Neukirch-Uchida theorem in algebraic number theory, can be reformulated as structure theorems for commensurators of certain profinite groups.
The paper forms a systematic survey on fundamental aspects of commensurators of profinite groups and may serve as a springboard for further research on totally disconnected locally compact groups.