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Shalom, Yehuda

Bounded generation and Kazhdan’s property (T). (English) Zbl 0980.22017

Publ. Math., Inst. Hautes Étud. Sci. 90, 145-168 (1999).
A group \(G\) is said to be boundedly generated if it has a finite subset \(S\) and some number \(\nu\), depending only on \(G\) and \(S\), such that every \(g\in G\) can be written as a product: \(g=g^{k_1}_1 g_2^{k_2} \dots g_\nu^{k_\nu}\), with \(g_i\in S\) and \(k_i\) integers. Let \(G\) be a topological group, \(K\subset G\) a subset, \(\varepsilon >0\), and \((\pi,H)\) a continuous unitary \(G\)-representation. A vector \(v\in H\) is called \((K,\varepsilon)\)-invariant, if \(\|\pi (g)v-v \|< \varepsilon\|v\|\) \(\forall g\in K\). The group \(G\) is said to have Kazhdan’s property \((T)\) if there exist a compact set \(K\subset G\) and \(\varepsilon>0\), such that every continuous unitary \(G\)-representation with a \((K,\varepsilon)\)-invariant vector contains a nonzero \(G\)-invariant vector. In this case, \((K,\varepsilon)\) are called Kazhdan constants for \(G\). In this paper the author makes strong use of the property of bounded generation in the study of Kazhdan’s property \((T)\).
Reviewer: Udai Tewari (Kanpur)

Cited in 1 Review
Cited in 69 Documents

MSC:

22E46 Semisimple Lie groups and their representations
22E50 Representations of Lie and linear algebraic groups over local fields
20G05 Representation theory for linear algebraic groups

Keywords:

unitary \(G\)-representation; topological group; Kazhdan constants; bounded generation; Kazhdan’s property
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References:

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