Received 18 August 2014; revised 16 September 2014; accepted 6 October 2014

ABSTRACT

In this paper, we investigate Kazhdan’s relative Property (T) for pairs, where is a topological group and is any subset of. We show that the pair has Property (FH) and every function conditionally of negative type on is X-bounded if the pair has relative Property (T). We also prove that has Property (T) when is a s-compact locally compact group generated by its subgroups and the pair has relative Property (T) for all.

Keywords:

Relative Property (T) of Topological Group, Function Conditionally of Negative Type, Property (FH)

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.

References