Example: is quasi-isometric to 1 if and only if
is finite.
Definition: A metric space is proper if closed balls of finite radius in
are compact. The action of a group
on a metric space
is cocompact if
is compact in the quotient topology.
The Švarc-Milnor Lemma: Let be a proper geodesic metric space. Let
act cocompactly and properly discontinuously on
. (Properly discontinuously means that for all compact
.) Then
is finitely generated and, for any
, the map
is a quasi-isometry (where is equipped with the word metric).
Proof: We may assume that is infinite and
is non-compact. Let
be large enough that the
-translates of
cover
. Set
Let . Let
. We want to prove that:
(a) generates,
(b) ,
(c) , there exists
such that
Note: and
.
(c) is obvious.
(b-i) is also obvious.
To complete the proof we need to show (a) and (b-ii).
Assume . Let
be such that
As ,
. Choose
, such that
and
for each
. Choose
such that
for each
. Let
, so
. Now
So, . Therefore
generates
.
Also,
as required.
Corollary: If is a finite index subgroup of a finitely generated group then
is quasi-isometric to
.
Two groups and
are commensurable if they have isomorphic subgroups of finite index. Clearly, if
and
are commensurable then they are quasi-isometric.
Example: .
Semidirect product is taken over the matrix This means that
, but
Let with eigenvalues
with
. Let
.
sits inside
as a uniform lattice, meaning
is a compact space.
Exercise 11: What is this quotient?
So, is a quasi-isomorphic to
But, Bridson-Gersten showed that
and
are commensurable if and only if the corresponding eigenvalues
have a common power.
Exercise 12: Let be the infinite regular
valent tree. Prove that for all
,
is quasi-isometric to
.
3 comments
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9 February 2009 at 9.08 pm
Henry Wilton
Thanks, Allison! I edited a couple of things, including giving Švarc his hacek.
9 February 2009 at 9.18 pm
metaficionado
Thank you for fixing the hacek and the matrix.
10 February 2009 at 9.19 am
Henry Wilton
I should also just mention that the proof of the Švarc-Milnor Lemma is the one given by de la Harpe in his Topics in Geometric Group Theory.