$3$-Manifold Groups are Virtually Residually $p$

Let f : M → N be a continuous map between closed irreducible graph manifolds with infinite fundamental group. Perron and Shalen (1999) [16] showed that if f induces a homology equivalence on all finite covers, then f is in fact homotopic to a homeomorphism. Their proof used the statement that every graph manifold is finitely covered by a 3-manifold whose fundamental group is residually p for every prime p. We will show that this statement regarding graph manifold groups is not true in general, but we will show how to modify the argument of Perron and Shalen to recover their main result. As a by-product we will determine all semidirect products Z Z d which are residually p for every prime p.

show abstract

“…We first prove (1). Since Im{ f * : π 1 (T ) → π 1 (N)} ∼ = Z it follows that the map T → N factors up to homotopy through a circle.…”

Section: ) For a Component A Of M As Inmentioning

confidence: 90%

“…In [1] we show that if N is a Seifert fibered 3-manifold, then π 1 (N) has a finite-index subgroup which is residually p for every p. In [1] we furthermore prove the following weaker statement regarding the fundamental groups of graph manifolds. Theorem 1.2.…”

Section: ) G Is Residually P For All P If and Only If G Is Nilpotentmentioning

confidence: 91%

Residual properties of graph manifold groups

Aschenbrenner

Friedl

2011

Topology and its Applications

Self Cite

View full text Add to dashboard Cite

show abstract

“…The reader may compare the hypotheses of Theorem 6.1 to the notion of Pefficiency (see [5] and [44]).…”

Section: 2mentioning

confidence: 99%

Residually finite rationally p groups

Koberda

Suciu

2019

Commun. Contemp. Math.

View full text Add to dashboard Cite

In this article we develop the theory of residually finite rationally p (RFRp) groups, where p is a prime. We first prove a series of results about the structure of finitely generated RFRp groups (either for a single prime p, or for infinitely many primes), including torsion-freeness, a Tits alternative, and a restriction on the BNS invariant. Furthermore, we show that many groups which occur naturally in group theory, algebraic geometry, and in 3-manifold topology enjoy this residual property. We then prove a combination theorem for RFRp groups, which we use to study the boundary manifolds of algebraic curves CP 2 and in C 2 . We show that boundary manifolds of a large class of curves in C 2 (which includes all line arrangements) have RFRp fundamental groups, whereas boundary manifolds of curves in CP 2 may fail to do so.

show abstract

“…For a graph of finite p-groups G = (X, G • ), properness is equivalent both to the existence of a finite quotient of Π 1 (G) into which all G x inject and to the property that the discrete fundamental group π 1 (G) (that is, the fundamental group when considered as a graph of discrete groups) is residually p. There are classical criteria for this property in the case of one-edge graphs of groups [Hig64,Cha94]. Criteria for more general graphs of groups also exist [AF13,Wil18].…”

Section: Properness Of Graphs Of Pro-p Groupsmentioning

confidence: 99%