1. Introduction
In the mid 60’s, Kazhdan defined the following Property (T) for locally compact groups and used this to prove that a large class of lattices are finitely generated.
Definition 1.1. [1] A topological group
has Property (T) if there exist a compact subset
and a real number
such that, whenever
is a continuous unitary representation of
on a Hilbert space ![]()
for which there exists a vector
of norm 1 with
, then there exists an invariant
vector, namely a vector
in
such that
for all
.
Since then, Property (T) has been studied extensively and there are a lot of publications. One can see [2] - [5] .
The notion of relative Property (T) for a pair
, where
is a normal subgroup of
, was implicit in Kazhdan’s paper [5] , and later made explicit by Margulis [6] . The definition of relative Property (T) has been extended in [7] to pairs
with
not necessarily normal in
. In order to obtain more information about the unitary dual of locally compact group, Cornulier in [3] extended the definition of relative Property (T) to pairs
, where
is any subset of the locally compact group
.
We know that if the pair
has relative Property (T), then there exists an open, compactly generated subgroup
of
, containing
, such that
has relative Property (T). Shalom [8] generalizes Kazhdan’s definition of Property (T) to topological groups that are not locally compact. There are many natural examples such as the loop group of all continuous functions from a circle to
and the pair
(
is a subgroup of
), which both have Property (T).
Inspired by the work of Cornulier and Shalom, in this paper, we go further in this direction and try to extend Cornulier’s result from locally compact groups to topological groups. The motivation for this is that, given a topological group
, the knowledge of the family of subsets X such that has relative Property (T) contains much more precise information than the bare information whether G has Property (T). On the other hand, relative Property (T) for topological groups is very important to discuss Haagerup Property.
2. Preliminaries
We first introduce some of basic notations and terminologies, the details can be found in [1] .
Definition 2.1. [1] Let
be a topological group, and
be a closed subgroup. The pair
has Property (T) if there exist a compact subset
and a real number
such that, whenever
is a continuous unitary representation of
on a Hilbert space
for which there exists a vector
of norm
1 with
, then there exists an invariant vector, namely a vector
in
such that
for all
.
Definition 2.2. Let
be an orthogonal representation of the topological group
on a real Hilbert space
.
1) A continuous mapping
such that
, for all
is called a 1- cocycle with respect to
.
2) A 1-cocycle
for which there exists
such that
, for all
is called a 1-coboudary with respect to
.
3) The space
of all 1-cocycles with respect to
is a real vector space under the pointwise operations, and the set
of all 1-coboundaries is a subspace of
. The quotient vector space
is called the first cohomology group with coefficients in
.
4) Let
. The affine isometric action associated to a cocycle
is the affine isometric action
of
on
defined by
![]()
where
is the canonical affine Hilbert space associated with ![]()
Definition 2.3. A continuous real valued kernel
on a topological space
is conditionally of negative
type if
, for all
, and
, for any elements
in
, and any real numbers
with
A continuous real value function
on a topological group
is conditionally of negative type if the kernel on
defined by
is conditionally of negative type.
Example 2.4. Let
be a topological group, and let
be an affine isometric action of
on a real Hilbert space
, according to Example C. 2.2 ii in [1] , for any
, the function
![]()
is conditionally of negative type.
In particular, for any orthogonal representation
on
and for any
, the function
is conditionally of negative type.
Theorem 2.5. [1] Let
be an orthogonal representation of the topological group G on a real Hilbert space
. Let
, with associated affine isometric action
. The following statements are equivalent:
1)
is bounded;
2) all the orbits of
are bounded;
3) some orbit of
is bounded;
4)
has a fixed point in
.
Definition 2.6. A topological group
has Property (FH) if every affine isometric action of
on a real Hilbert space has a fixed point. Let
be a closed subgroup of
. The pair
has Property (FH) if every affine isometric action of
on a real Hilbert space has an
-fixed point.
The following theorem describes connection among bounded functions conditionally of negative type, Property (FH) and cohomology groups pair
.
Theorem 2.7. [1] Let
be a topological group,
be a closed subgroup of
. The following statements are equivalent:
1)
is the zero mapping, for every orthogonal representation
of
,
2)
has Property (FH),
3) every function conditionally of negative type on
is H-bounded.
Theorem 2.8. [1]
be a topological group, H be a closed subgroup of
.
1) If Pair
has relative Property (T), then pair
has Property (FH).
2) If Pair
has relative Property (T), then every function conditionally of negative type on
is
bounded.
3) Assume that
is a
-compact locally compact group and that
has Property (FH), then pair
has relative Property (T).
Proof. By virtue of [1] Remark 2.12.5: if the pair
has relative Property (T), then
has Property (FH) and Theorem 2.7 1), 2) is obvious and 3) is the Delorme-Guichardet Theorem applied to the pair
, consisting of a group
and a subgroup
(see [1] , Exercise 2.14.9). ![]()
Theorem 2.9. [4] Let
be a topological group and
be subgroups of
such that
is generated by
. Each pair
has relative Property (T). Then G has Property (T).
3. Relative Property (T) of Pairs for Topological Groups and Subsets
When
be a locally compact group, Cornulier extended the definition of relative Property (T) to pairs
, where
is any subset of
, and then established various characterizations of relative Property (T) for the pair
, which were already known in [1] [9] for the case that
is a topological group,
is a closed subgroup. We extend the definition of relative Property (T) to pairs
, where
is a topological group,
is any subset of
, and then established similar characterizations of relative Property (T) for the pair
.
Definition 3.1. Let
be a topological group, and
is any subset of G. The pair
has relative Property (T) if there exist a compact subset
and a real number
such that, whenever
is a continuous unitary representation of
on a Hilbert space
for which there exists a vector
of norm
1 with
, then there exists an invariant vector, namely a vector
in
such that
for all
.
Definition 3.2. Let
be a topological group, and
is any subset of
. The pair
has Property (FH) if every affine isometric action of
on a real Hilbert space has an
-fixed point.
The following theorem will establish some characterizations of relative Property (T) for a pair
.
Theorem 3.3. Let
be a topological group and
be any subset of
.
1) If pair
has relative Property (T), then pair
has Property (FH).
2) If pair
has relative Property (T), every function conditionally of negative type on
is
- bounded.
3) Assume that
is a
-compact locally compact group and that pair
has Property (FH), then the pair
has relative Property (T).
We will apply the following lemma to prove above theorem:
Lemma 3.4. Let
be a topological group,
be any subset and
be a close subgroup of
, where
generates
. Then the pair
has relative Property (T) if and only if the pair
has relative Property (T).
Proof. Necessity. Since the pair
has relative Property (T), there exist a compact subset
and a real number
such that whenever
is a continuous unitary representation of
on a Hilbert space
for which there exists a vector
of norm 1 with
, then there exists an invariant vector
in
such that
for all
. Hence
1)
.
2) Since
,
.
3)
.
4) According to
is a continuous unitary representation of
on a Hilbert space
, if
, where
, then
.
According to 1), 2), 3) and 4), pair
has relative Property (T).
Sufficiency. It follows from definition of relative Property (T).
Proof of Theorem 3.3. According to Lemma 3.4, the pair
has relative Property (T), where
generate
, then by Theorem 2.8, every function conditionally of negative type on
is H-bounded.
1) By Theorem 2.7, the pair
has Property (FH), hence the pair
has Property (FH).
2) Since every function conditionally of negative type on
is
-bounded, it is
-bounded too.
3) According to [3] (Theorem 1.1), the result holds. ![]()
Example 3.5 Let K be a any non-archimedean local field, we consider the following subgroups of
:
![]()
![]()
Then the pair
,
do not have relative Property (T) .
In fact, if
has relative Property (T), then
has relative Property. Since the group
is generated by
, by Theorem 2.9,
has Property (T). This contradicts the fact that
does not have Property (T). By Lemma 3.4,
do not have relative Property (T).
Remark 3.6. By Theorem 3.3,
does not have Property (FH).
do not have relative Property (FH). According to [1] (remark: 1.4.4 (iii) on Page 47), groups
do not have Property (T), hence do not have Property (FH).
Theorem 3.7. Let
be a topological group,
be subsets. Denote by
the point- wise product
. Suppose that, for every
,
has relative Property (T). Then pair
has Property (FH).
Proof. It suffices to prove the case when n = 2, since the result follows by induction on
. By Theorem 3.3,
has Property (FH), and every function conditionally of negative type on
is
-bounded. Let ![]()
be any conditionally negative definite function on G. According to the inequality,
,
the pair
is
-bounded. Hence, pair
is
-bounded.
Remark 3.8. In [3] , Cornulier show when
be a locally compact group for above theorem, then pair
has relative Property (T).
Theorem 3.9. Let
be a topological group,
be subsets. Suppose that, for every
,
has relative Property (T), then we have:
1) the pair
has Property (FH).
2) if
, then the pair
has Property (FH).
Proof. 1)
, then there exists a
such that
. Since pair
has relative Property
(T), let
be any function conditionally of negative type on
, then
, that is,
is an
- bounded function. Similarly, we can easily prove result 2). ![]()
Theorem 3.10. Let
be a topological group.
1) Let
be a subgroup of
,
. If the pair
has relative Property (T), then
has relative Property (T).
2) Let
be a subgroup of
,
. If the pair
has Property (FH), then
has Property (FH).
3) Let
be subgroups of
,
. Let
be a topological group generated by
. If
has relative Property (T) for every
, then the pair
has Property (FH).
Proof. 1) Since
has relative Property (T), if there exist a compact subset
and a real number
such that,
is a continuous unitary representation of
on a Hilbert space
, then
is a continuous unitary representation of
on a Hilbert space
for which there exists a vector
of norm
1 with
, there exists an invariant vector, namely a vector
in
such that
for all
. According to
,
for all
. Hence
has relative Property (T).
2) Since
has Property (FH), every affine isometric action of
on a real Hilbert space is an affine isometric action of
, and thus it has an
-fixed point. Hence, it has an
-fixed point. That is,
has Property (FH).
3) It suffices to prove the case when n = 2, since then the result follows by induction. According to 1) and Theorem 3.9,
, for every
, the pair
has relative Property (T), then the pair
has Property (FH). By using the result in 2), the pair
has Property (FH). ![]()
In [4] , M.Ershov and A. Jaikin-Zapirain showed the following facts: Suppose G be a group generated by subgroups
, and the pair
has relative Property (T) for each i, and any two subgroups ![]()
and
are
-orthogonal for some
. Then G has Property (T).
We will naturally ask: Can we remove the hypothesis “any two subgroups
and
are
-orthogonal for some
”?
The following Theorem solves the above problem partially:
Theorem 3.11. Let
be a
-compact locally compact group generated by subgroups
. Suppose that the pair
has relative Property (T) for each
. Then
has Property (T).
Proof. According to Theorem 3.9, the pair
has Property (FH). Then by Theorem 3.3, the pair
has relative Property (T). Since
is a group generated by subsets
, according to Lemma 3.4, the pair
has relative Property (T), then
has Property (T). ![]()
Acknowledgements
The work is finished during the first author visiting Auburn University. He thanks Professor Tin-Yau Tam for his invitation to visit Auburn University and participate the Linear Algebra Seminar.