Margulis Lemma

STRUCTURE OF FUNDAMENTAL GROUPS 7
Proof. We choose a measurable section j : B 1 (p) → B 1 (˜p), i.e. π(j(x)) = x for any
x ∈ B 1 (p). Let T = j(B 1 (p)). Then we obviously have that diam(T ) ≤ 2 and
∫ ∫
− f = − ˜f.
B 1(p)
Let S be the union of g(T ) over all g ∈ π 1 (M) such that g(T ) ∩ B 1 (˜p) ≠ ∅. It’s
obvious from the triangle inequality that S ⊂ B 3 (˜p). It is also clear that
∫ ∫
− f = − ˜f
B 1(p)
and B 1 (˜p) ⊂ S. Lastly, notice that by Bishop–Gromov relative volume comparison
vol B 3 (˜p) ≤ C(n) vol B 1 (˜p) for some universal constant C(n). Since B 1 (˜p) ⊂ S ⊂
B 3 (˜p), we also have vol B 3 (˜p) ≤ C(n) vol S and hence
∫
∫ ∫
− ˜f ≤ C(n)− ˜f = C(n)− f.
B 1(˜p)
S
T
S
B 1(p)
It is easy to see that the above proof generalizes to an arbitrary cover of M.
1.3. Equivariant Gromov–Hausdorff convergence. Let Γ i be a sequence of
closed subgroups of the isometry groups of metric spaces X i and ˜p i ∈ X i . If
(X i , ˜p i ) converges in the pointed Gromov–Hausdorff topology to (Y, ˜p ∞ ), then after
passing to a subsequence one can find a closed subgroup G ⊂ Iso(Y ) such that
(X i , Γ i , ˜p i ) → (Y, G, ˜p ∞ ) in the equivariant Gromov–Hausdorff topology. For details
we refer the reader to [FY92].
Definition 1.7. A sequence of subgroups Υ i ⊂ Γ i is called uniformly open, if there
is some ε > 0 such that Υ i contains the set
{
g ∈ Γi | d(gq, q) < ε for all q ∈ B 1/ε (˜p i ) } for all i.
Γ i is called uniformly discrete if {e} ⊂ Γ i is uniformly open. We say the sequence
Υ i is boundedly generated if there is some R such that Υ i is generated by
{g ∈ Υ i | d(g˜p i , ˜p i ) < R}.
Lemma 1.8. Let Υ j i ⊂ Γ i be uniformly open with Υ j i → Υj ∞ ⊂ G, j = 1, 2.
a) Υ 1 i ∩ Υ2 i is uniformly open and converges to Υ1 ∞ ∩ Υ 2 ∞.
b) If g i ∈ Γ i converges to g ∞ ∈ G then g i Υ 1 i g−1 i is uniformly open and converges
to g ∞ Υ 1 ∞g∞ −1 .
c) Suppose in addition that Υ j i is boundedly generated, j = 1, 2. If Υ1 ∞ ∩ Υ 2 ∞
has finite index H in Υ 1 ∞, then Υ 1 i ∩ Υ2 i has index H in Υ1 i for all large i.
The proof is an easy exercise.
Example 1 (Z l actions converging to a Z 2 -action.). Consider a 2-torus T 2 k
given by
a Riemannian product S 1 ×S 1 where each of the factors has length k. Put l = k 2 +1
and let Z/lZ act on T 2 k by ( e 2πi
k 2 k2πi )
+1 , e k 2 +1 . In the equivariant Gromov–Hausdorff
topology the action will converge (for k → ∞) to the standard Z 2 action on R 2 .
One can, of course, define a similar sequence of actions on S 3 × S 3 but in this
case the actions will not be uniformly cocompact.
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