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

Abstract

There has been much recent interest into those properties of a 3manifold 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. Z = lim ← − Z/n of all finite cyclic groups. The pro-π completion, denoted Z π , is the inverse limit of those finite cyclic groups whose orders only involve primes from π. Because of the Chinese Remainder Theorem, these finite cyclic groups split as products of cyclic groups of prime power order, and these splittings are natural with respect to the quotient maps Z/mn → Z/n. It follows that the profinite completion of Z splits as the direct product, over all primes p, of the rings of p-adic integersThis splitting as a direct product, which of course is not a feature of Z itself, means that ring-theoreticallyẐ behaves less well than Z. In particular, there

Pro-p subgroups of profinite completions of 3-manifold groups

“…This was stated in [, p. 376], and a careful proof was written down in [, Proposition 6.8]. Proposition follows in a straightforward manner from the following statement.…”

Profinite rigidity of graph manifolds, II: knots and mapping classes

“…In the paper [Wil18] the author proved a result classifying graph manifolds by the profinite completions of their fundamental groups. This classification (see Theorem 1.6 of the present paper) takes the form of a finite list of numerical conditions which are easy to check for a given pair of graph manifolds when presented in a certain standard form.…”

“…This classification (see Theorem 1.6 of the present paper) takes the form of a finite list of numerical conditions which are easy to check for a given pair of graph manifolds when presented in a certain standard form. This present paper represents a continuation of [Wil18]. A certain familiarity with at least Section 10 of [Wil18] will be required.…”

“…This present paper represents a continuation of [Wil18]. A certain familiarity with at least Section 10 of [Wil18] will be required. Initially we seek to shed light on the classification theorem by means of examples.…”

“…Following this we will use the techniques from [Wil18] to investigate two other entities of great interest to low dimensional topologists-knots and mapping classes. In Section 4 we will study those knot exteriors in S 3 which are graph manifolds and prove that they are all determined by the profinite completions of their fundamental groups.…”