Read e-book online 3-manifold groups are virtually residually p PDF

By Matthias Aschenbrenner;Stefan Friedl

ISBN-10: 0821888013

ISBN-13: 9780821888018

Given a primary $p$, a gaggle is termed residually $p$ if the intersection of its $p$-power index basic subgroups is trivial. a bunch is termed almost residually $p$ if it has a finite index subgroup that is residually $p$. it's recognized that finitely generated linear teams over fields of attribute 0 are almost residually $p$ for all yet finitely many $p$. particularly, basic teams of hyperbolic $3$-manifolds are almost residually $p$. it's also recognized that primary teams of $3$-manifolds are residually finite. during this paper the authors end up a standard generalisation of those effects: each $3$-manifold crew is almost residually $p$ for all yet finitely many $p$. this offers proof for the conjecture (Thurston) that primary teams of $3$-manifolds are linear teams

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Moreover, the filtration of ω determined by {Dn } is precisely {ω n }, the powers of the augmentation ideal. The ideal ω is nilpotent of nilpotency class (p − 1) n dimFp (Dn /Dn+1 ). 7]. The nilpotency class of an ideal I of R[G] is the smallest d ≥ 0, if it exists, such that I d+1 = 0. By the theorem, the nilpotency class of ω is an upper bound on the lower p-length (and hence the nilpotency class) of G. We denote by ann(ω) := {r ∈ R : rω = 0} 32 2. EMBEDDING THEOREMS FOR p-GROUPS the left annihilator of ω (a left ideal of R[G]).

H (g)) ∈ Similarly ones sees that each edge group of G is closed in G. 1) is injective; hence if in addition the fundamental group of G/Hi is residually P for i = 1, 2, then so is the fundamental group of G/(H1 ∩ H2 ). 11. Suppose that Ge = 1 for every e ∈ E(Y ). If G satisfies condition (A) in the previous proposition, then G is residually P. 12. The fundamental group of every graph of groups whose vertex groups are residually P and whose edge groups are trivial is residually P. Proof. By the previous corollary, it is enough to show that if v0 ∈ V (Y ) is such that Gv0 is a P-group and Gv = 1 for all v ∈ V (Y ) with v = v0 , and Ge = 1 for all e ∈ E(Y ), then G is residually P.

Then G is residually P. Proof. In this proof H ranges over all compatible collections of normal subgroups such that π1 (G/H) is residually P. The natural morphisms πH : π1 (G) → π1 (G/H) combine to a morphism π : G → G∗ := π1 (G/H). H Since G∗ is residually P, it is enough to show that π is injective. Let g = 1 be an element of G, and represent g by a path γ = (g0 , e1 , g1 , e2 , . . 2. Then πH (g) is represented by the path πH (γ) := (πH (g0 ), e1 , πH (g1 ), e2 , . . , en , πH (gn )) 46 3.

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3-manifold groups are virtually residually p by Matthias Aschenbrenner;Stefan Friedl

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