Inter-universal Teichmuller Theory I: Construction of Hodge Theaters

INTER-UNIVERSAL TEICHMÜLLER THEORY I 47
Finally, we observe that Proposition 2.4, Corollary 2.5 admit the following
discrete analogues, which may be regarded as generalizations of [André], Lemma
3.2.1 [cf. Theorem 2.6 below in the case where H = F = G is free]; [EtTh], Lemma
2.17.
Theorem 2.6. (Profinite Conjugates of Discrete Subgroups) Let F be
a group that contains a subgroup of finite index G ⊆ F such that G is either a
free discrete group of finite rank or an orientable surface group [i.e., a
fundamental group of a compact orientable topological surface of genus ≥ 2]; H ⊆ F
an infinite subgroup. Since F is residually finite [cf., e.g., [Config], Proposition 7.1,
(ii)], we shall write H, G ⊆ F ⊆ ̂F ,where ̂F denotes the profinite completion of
F .Letγ ∈ ̂F be an element such that
γ · H · γ −1 ⊆ F [or, equivalently, H ⊆ γ −1 · F · γ].
Write H G
def
= H ⋂ G. Then γ ∈ F · N̂F
(H G ), i.e., γ · H G · γ −1 = δ · H G · δ −1 ,for
some δ ∈ F . If, moreover, H G is nonabelian, thenγ ∈ F .
Proof. Let us first consider the case where H G is abelian. In this case, it follows
from Lemma 2.7, (iv), below, that H G is cyclic. Thus, by applying Lemma 2.7,
(ii), it follows that by replacing G by an appropriate finite index subgroup of G,
we may assume that the natural composite homomorphism H G ↩→ G ↠ G ab is a
split injection. In particular, by Lemma 2.7, (v), we conclude that NĜ(H G )=ĤG,
wherewewriteĤG for the closure of H G in the profinite completion Ĝ of G. Next,
let us observe that by multiplying γ on the left by an appropriate element of F ,we
may assume that γ ∈ Ĝ. Thus,wehaveγ · H G · γ −1 ⊆ F ⋂ Ĝ = G. Next,letus
recall that G is conjugacy separable. Indeed, this is precisely the content of [Stb1],
Theorem 1, when G is free; [Stb2], Theorem 3.3, when G is an orientable surface
group. SinceG is conjugacy separable, it follows that γ · H G · γ −1 = ɛ · H G · ɛ −1 for
some ɛ ∈ G, soγ ∈ G·NĜ(H G )=G·ĤG ⊆ F (H ·N̂F G ), as desired. This completes
the proof of Theorem 2.6 when H G is abelian.
Thus, let us assume for the remainder of the proof of Theorem 2.6 that H G is
nonabelian. Then, by applying Lemma 2.7, (iii), it follows that, after replacing G by
an appropriate finite index subgroup of G, we may assume that there exist elements
x, y ∈ H G that generate a free abelian subgroup of rank two M ⊆ G ab such that
the injection M↩→ G ab splits. Write H x ,H y ⊆ H G for the subgroups generated,
respectively, by x and y; Ĥ x , Ĥy ⊆ Ĝ for the respective closures of H x, H y . Then
by Lemma 2.7, (v), we conclude that NĜ(H x )=Ĥx, NĜ(H y )=Ĥy. Next, let us
observe that by multiplying γ on the left by an appropriate element of F ,wemay
assume that γ ∈ Ĝ. Thus, we have γ · H G · γ −1 ⊆ F ⋂ Ĝ = G. In particular, by
applying the portion of Theorem 2.6 that has already been proven to the subgroups
H x ,H y ⊆ G, we conclude that γ ∈ G · NĜ(H x )=G · Ĥx, γ ∈ G · NĜ(H y )=G · Ĥy.
Thus, by projecting to Ĝab , and applying the fact that M is of rank two, we conclude
that γ ∈ G, as desired. This completes the proof of Theorem 2.6. ○
Remark 2.6.1. Note that in the situation of Theorem 2.6, if H G is abelian, then
— unlike the tempered case discussed in Proposition 2.4! — it is not necessarily
thecasethatF = γ −1 · F · γ.