Margulis Lemma

30 VITALI KAPOVITCH AND BURKHARD WILKING
Thus, the normalizer of Υ i has finite index ≤ [G : G ′ ] in Γ i for all large i and
Step 4 is established.
Step 5. The desired contradiction arises if, after passing to a subsequence, the
indices [Γ i : Γ iε ] ∈ N ∪ {∞} converge to ∞.
By Step 4 there is a subgroup Υ i ⊂ Γ iε of index ≤ H which is normalized by a
subgroup of Γ i of index ≤ H. Similarly to the beginning of the proof of Step 3, one
can reduce the situation to the case of Υ i ⊳ Γ i .
Moreover, Υ i ⊳ Γ i is uniformly open and hence the action of Γ i /Υ i on M i /Υ i is
uniformly discrete and converges to a properly discontinuous action of the virtually
abelian group G/Υ ∞ on the space R k × (R l × ˜K)/Υ ∞ . It is now easy to see that
Γ iε /Υ i contains an abelian subgroup of controlled finite index for large i.
After replacing Γ i once more by a subgroup of controlled finite index we may
assume that Γ i /Υ i is abelian. This also shows that without loss of generality
Υ i = Γ iε .
Recall that by Step 2 we have assumed that f j i converges to the identity in the
weakly measured sense for every j = 1, . . . , k. Therefore, the same is true for the
commutator [f j i , γ i] ∈ Γ i if γ i ∈ Γ i has bounded displacement. This in turn implies
that [f j i , γ i] ∈ Γ iε for all large i and j = 1, . . . , k.
By Theorem 2.5, we can find for some large σ elements d i 1, . . . , d i σ ∈ Γ i with
bounded displacement generating Γ i . We also assume that the first τ elements
generate Γ iε for some τ < σ.
Let ˆΓ i := 〈d i 1, . . . , d i σ−1〉. Note that ˆΓ i ⊳ Γ i since ˆΓ i contains Γ iε = Υ i ⊳ Γ i and
Γ i /Γ iε is abelian.
If for some subsequence and some integer H the group ˆΓ i has index ≤ H in Γ i
then we may replace Γ i by ˆΓ i . Thus, without loss of generality, the order of the
cyclic group Γ i /ˆΓ i tends to infinity. The element f k+1
i := d i σ ∈ Γ i represents a
generator of this factor group which has bounded displacement.
We have seen above that f j i normalizes ˆΓ i and [f k+1
i , f j i ] ∈ Γ iε ⊂ ˆΓ i , j = 1, . . . , k.
Clearly f k+1
i , being an isometry with bounded displacement, has the zooming in
property.
We replace M i by B i := ˜M i /ˆΓ i . Notice that (B i , ˆp i ) converges to (R k × (R l ×
˜K)/Ĝ, ˆp ∞) where Ĝ is the limit group of ˆΓi . Since G/Ĝ is a noncompact group
acting cocompactly on (R l × ˜K)/Ĝ, we see that (Rl × ˜K)/Ĝ splits as Rp × K ′ with
K ′ compact and p > 0. If p > 1, we put f j i = id for j = k + 2, . . . , k + p.
Thus the induction assumption applies: There is a positive integer C such that
the following holds for all large i:
We can find a subgroup N i of π 1 (B i ) of index ≤ C and a nilpotent chain of
normal subgroups {e} = N 0 ⊳ . . . ⊳ N n−k−1 = N i such that the quotients are cyclic.
Moreover, the groups are invariant under the automorphism induced by conjugation
of (f j i )C! and the induced automorphism of N h+1 /N h is the identity, j = 1, . . . , k+1.
We put d = C!, and consider the group ¯N i ⊂ π 1 (M i ) generated by (f k+1
i ) d and
N i . Clearly, the index of ¯N i in π 1 (M i ) is bounded by C · d.
Moreover, the chain {e} = N 0 ⊳ . . . ⊳ N n−k−1 ⊳ N n−k := ¯N i is normalized by the
elements (f j i )d (j = 1, . . . , k) and the action on the cyclic quotients is trivial.
In other words, the sequence fulfills the conclusion of our theorem – a contradiction.
□