Abstract commensurators of profinite groups

Details Abstract commensurators of profinite groups Abstract commensurators of profinite groups

2008-10-11

arxiv.org/abs/0810.2060v2

Trans. Amer. Math. Soc. 363 (2011), no. 10, 5381--5417

Mathematics

Group Theory

37 pages, final version

Scientific paper

10.1090/S0002-9947-2011-05295-5

In this paper we initiate a systematic study of the abstract commensurators of profinite groups. The abstract commensurator of a profinite group $G$ is a group $Comm(G)$ which depends only on the commensurability class of $G$. We study various properties of $Comm(G)$; in particular, we find two natural ways to turn it into a topological group. We also use $Comm(G)$ to study topological groups which contain $G$ as an open subgroup (all such groups are totally disconnected and locally compact). For instance, we construct a topologically simple group which contains the pro-2 completion of the Grigorchuk group as an open subgroup. On the other hand, we show that some profinite groups cannot be embedded as open subgroups of compactly generated topologically simple groups. Several celebrated rigidity theorems, like Pink's analogue of Mostow's strong rigidity theorem for simple algebraic groups defined over local fields and the Neukirch-Uchida theorem, can be reformulated as structure theorems for the commensurators of certain profinite groups.

Affiliated with

Also associated with

No associations

Rating

Abstract commensurators of profinite groups does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Abstract commensurators of profinite groups, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Abstract commensurators of profinite groups will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-200214