By Matthias Aschenbrenner;Stefan Friedl
Given a first-rate $p$, a gaggle is named residually $p$ if the intersection of its $p$-power index common subgroups is trivial. a gaggle is named almost residually $p$ if it has a finite index subgroup that is residually $p$. it truly is recognized that finitely generated linear teams over fields of attribute 0 are nearly residually $p$ for all yet finitely many $p$. specifically, primary teams of hyperbolic $3$-manifolds are almost residually $p$. it's also recognized that basic 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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Extra resources for 3-manifold groups are virtually residually p
M, where m = length of K, deﬁne the normal subgroup Wi := Ki · T K = (k, f ) ∈ W : k ∈ Ki , f ∈ T K of W , and for i = 1, . . , l, where l = nilpotency class of W , set Wm+i := T K ∩ γip (W ). Then W = W1 ≥ W2 ≥ · · · ≥ Wm ≥ Wm+1 = T K ≥ Wm+2 ≥ · · · ≥ Wm+l ≥ 1 is a central p-ﬁltration of W . 2. Clearly α−1 (Wi ) = θ −1 (Ki ) = Gi Gn for 1 ≤ i ≤ n for n ≤ i ≤ m + 1. 2. 10, Wm+l ∩ X K = T K ∩ X[K] = X K = α(X), hence α−1 (Wm+l ) = X = Gn . Therefore α−1 (Wm+1 ) = α−1 (Wm+2 ) = · · · = α−1 (Wm+l ) = Gn .
R has p-power order: in this case, the semidirect product H = G∗ A has the right property. Note however that it is not enough to have, for each i individually, an extension of ϕi to a p-automorphism of some p-group containing G as a subgroup: Example. Suppose G = F2p , and consider the automorphisms ϕ = ( 10 11 ) and ψ = ( 11 01 ) of G. Then ϕ and ψ both have order p; however ϕ and ψ do not commute. 8]), and UT1 (2, Fp ) is abelian, there are no extensions of ϕ, ψ to automorphisms of a p-group containing G as a subgroup and which generate a p-group.
Then G is residually P. Digression: necessity of conditions (A) and (B). ) It is interesting to ask whether the conditions (A) and (B) in the previous proposition are necessary for G to be residually P. 4 we immediately obtain: 48 3. 15. If G is residually P, then G satisﬁes (A). The situation for condition (B) is slightly less clear. 16. Suppose for each edge e ∈ E(Y ), either (1) the subgroup fe (Ge ) of Gt(e) is maximal with respect to being verbal with respect to a collection We of words; or (2) Gt(e) is abelian and fe (Ge ), fe (Ge ) are proper subgroups of Gt(e) , Gt(e) , respectively.
3-manifold groups are virtually residually p by Matthias Aschenbrenner;Stefan Friedl