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Register a new userAbstract commensurators of profinite groups
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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 Pinks analogue of Mostows 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.
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Let $Gamma$ be the fundamental group of a surface of finite type and Comm$(Gamma)$ be its abstract commensurator. Then Comm$(Gamma)$ contains the solvable Baumslag--Solitar groups $langle a ,b : a b a^{-1} = b^n rangle$ for any $n > 1$. Moreover, the Baumslag--Solitar group $langle a ,b : a b^2 a^{-1} = b^3 rangle$ has an image in Comm$(Gamma)$ that is not residually finite. Our proofs are computer-assisted. Our results also illustrate that finitely-generated subgroups of Comm$(Gamma)$ are concrete objects amenable to computational methods. For example, we give a proof that $langle a ,b : a b^2 a^{-1} = b^3 rangle$ is not residually finite without the use of normal forms of HNN extensions.
We study the subgroup structure of the etale fundamental group $Pi$ of a projective curve over an algebraically closed field of characteristic 0. We obtain an analog of the diamond theorem for $Pi$. As a consequence we show that most normal subgroups of infinite index are semi-free. In particular every proper open subgroup of a normal subgroup of infinite index is semi-free.
We introduce and investigate a class of profinite groups defined via extensions of centralizers analogous to the extensively studied class of finitely generated fully residually free groups, that is, limit groups (in the sense of Z. Sela). From the fact that the profinite completion of limit groups belong to this class, results on their group-theoretical structure and homological properties are obtained.
The article deals with profinite groups in which the centralizers are pronilpotent (CN-groups). It is shown that such groups are virtually pronilpotent. More precisely, let G be a profinite CN-group, and let F be the maximal normal pronilpotent subgroup of G. It is shown that F is open and the structure of the finite quotient G/F is described in detail.
A group $G$ is said to have restricted centralizers if for each $g$ in $G$ the centralizer $C_G(g)$ either is finite or has finite index in $G$. Shalev showed that a profinite group with restricted centralizers is virtually abelian. Given a set of primes $pi$, we take interest in profinite groups with restricted centralizers of $pi$-elements. It is shown that such a profinite group has an open subgroup of the form $Ptimes Q$, where $P$ is an abelian pro-$pi$ subgroup and $Q$ is a pro-$pi$ subgroup. This significantly strengthens a result from our earlier paper.
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