Permanence criteria for semi-free profinite groups

虚拟满射与射影群的 $n$-$(n+1)$-$(n+2)$ 定理

Virtual Surjection and the $n$-$(n+1)$-$(n+2)$ Theorem for Profinite Groups

本文证明了射影群的虚拟满射猜想和 $n$-$(n+1)$-$(n+2)$ 猜想，并给出了纤维积的 $\mathrm{FP}_n$ 保持性结果，主要工具是射影群模的 $\mathrm{FP}_n$ 性质的数值判据。

Bieri-Eckmann Criteria for Profinite Groups

In this paper we derive necessary and sufficient homological and cohomological conditions for profinite groups and modules to be of type $\operatorname{FP}_n$ over a profinite ring $R$, analogous to the Bieri-Eckmann criteria for abstract groups. We use these to prove that the class of groups of type $\operatorname{FP}_n$ is closed under extensions, quotients by subgroups of type $\operatorname{FP}_n$, proper amalgamated free products and proper $\operatorname{HNN}$-extensions, for each $n$. We show, as a consequence of this, that elementary amenable profinite groups of finite rank are of type…

Inverse system characterizations of the (hereditarily) just infinite property in profinite groups

We give criteria on an inverse system of finite groups that ensure the limit is just infinite or hereditarily just infinite. More significantly, these criteria are 'universal' in that all (hereditarily) just infinite profinite groups arise as limits of the specified form. This is a corrected and revised version of the article: 'Inverse system characterizations of the (hereditarily) just infinite property in profinite groups', Bull. LMS vol 44, 3 (2012) 413--425.

On (hereditarily) just infinite profinite groups that are not virtually pro-p

A profinite group G is just infinite if every non-trivial closed normal subgroup of G is of finite index, and hereditarily just infinite if every open subgroup is just infinite. Hereditarily just infinite profinite groups need not be virtually pro-p, as shown in a recent paper of Wilson. The same paper gives a criterion on an inverse system of finite groups that is sufficient to ensure the limit is either virtually abelian or hereditarily just infinite. We give criteria of a similar nature that characterise the just infinite and hereditarily just infinite properties under the assumption that G…

Profinite rigidity of graph manifolds and JSJ decompositions of 3-manifolds

There has been much recent interest into those properties of a 3-manifold determined by the profinite completion of its fundamental group. In this paper we give readily computable criteria specifying precisely when two orientable graph manifold groups have isomorphic profinite completions. Our results also distinguish graph manifolds among the class of all 3-manifolds and give information about the structure of totally hyperbolic manifolds, and give control over the pro-$p$ completion of certain graph manifold groups.

On finite generation of self-similar groups of finite type

A self-similar group of finite type is the profinite group of all automorphisms of a regular rooted tree that locally around every vertex act as elements of a given finite group of allowed actions. We provide criteria for determining when a self-similar group of finite type is finite, level-transitive, or topologically finitely generated. Using these criteria and GAP computations we show that for the binary alphabet there is no infinite topologically finitely generated self-similar group given by patterns of depth $3$, and there are $32$ such groups for depth $4$.