NORMAL SUBGROUPS OF DIFFEOMORPHISM AND ...

on the rest of M both coincide with the identity map. Thus g = ḡ −1 f −1 ḡf
belongs to H since it is a conjugate of f −1 composed with f.
✷
If dim(M) = 1, so that M ≈ R with e = +∞, it is easy to show that H
contains a diffeomorphism f 1 that is strictly increasing near +∞. If B m+1
lies to the right of B m for every m, then f 1 can be conjugated to the desired
diffeomorphism f ∈ H.
We remark that the proof of Proposition ?? is similar to steps in the proof
of Lemma 2.7 of [?] and related to results in [?].
Proof of Theorem ??. First, we claim that h must move points that converge
to some end e of M. To see this, let K m be an increasing nested
sequence of compact sets such that K m ⊂ Int(K m+1 ) for every m and
∪ ∞ m=1K m = M. Choose a connected component U m of M K m for m =
1, 2, . . . , so that the closure of supp(h) ∩ U m is not compact, and so that
U m ⊃ U m+1 . Then the sets U m form a neighborhood base of a unique end e,
and h moves points arbitrarily close to e.
Now if the given diffeomorphism g has support in a ball, then by Proposition
??, g ∈ H. In general, if g ∈ A 0 (M) is any diffeomorphism isotopic to
the identity map by an isotopy with compact support, then by Corollary ??
of the Factorization Lemma, g is a composition of diffeomorphisms supported
in collared closed balls, each of them belonging to H, so again g ∈ H. ✷
Proof of Proposition ??. Let e be a product end of a manifold M. Suppose
that h ∈ A 0 A e so that supp(h) meets every neighborhood of e, and let
H be a normal subgroup of A 0 containing h. Take a product neighborhood
V 1
∼ = N×(0, 1] of the end e and identify V1 with N×(0, 1]. Then the sets V m =
N × (0, 1/m], m = 1, 2, . . . , form a basis of nested closed neighborhoods of
e. Let T : V 1 → V 2 be a diffeomorphism defined by setting T (x, s) = (x, t(s))
where t : (0, 1] → (0, 1/2] satisfies t(1/m) = 1/(m + 1) (for example, t(s) =
s
). Take a finite set B s+1 1, B 2 , . . . , B k of closed balls in Int(V 1 ) V 4 whose
interiors cover the compact set V 2 Int(V 3 ) and such that for j = 1, . . . , k
and m > 0, T m (B j ) ∩ B j = ∅. Now we may conjugate the diffeomorphism
f ∈ H given by Proposition ?? by an element of A 0 to a diffeomorphism
f j ∈ H supported in V 1 so that fj m (B j ) = T m (B j ) for m = 1, 2, . . . . Let
g j be a diffeomorphism supported on ∪ ∞ m=0T m (B j ) so that for each point
(x, s) ∈ Int(B j ), g j (x, s) = (x, s ′ ) for some s ′ = s ′ (x, s) < s and such that g j
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