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Strong property (T) for higher rank lattices. (English) Zbl 1429.22008

A length function on a locally compact topological group \(G\) is a function \(\ell : G \to \mathbb{R}^+\) such that (i) \(\ell\) is bounded on compact subsets of \(G\); (ii) \(\ell(g^{-1})=\ell(g)\) for every \(g \in G\); (iii) \(\ell(gh) \leq \ell(g)+\ell(h)\) for every \(g,h \in G\). The locally compact group \(G\) is said to have the (Lafforgue) strong property \((T)\) if for every length function \(\ell\), there exists \(s>0\) such that for every \(c>0\), the Banach algebra \(\mathcal{C}_{s\ell+c}(G)\) has a Kazhdan projection.
In the paper under review, the author proves that every lattice in a product of higher-rank simple Lie groups or higher-rank simple algebraic groups over local fields has the Lafforgue strong property (T).
A two-step representation of a topological group \(G\) is a tuple \((X_0,X_1,X_2,\pi_0,\pi_1)\) where \(X_0,X_1,X_2\) are Banach spaces and \(\pi_i: G \to B(X_i,X_{i+1})\) are strongly continuous maps such that \(\pi_1(gg') \pi_0(g'') = \pi_1(g) \pi_0(g' g'')\) for every \(g,g',g'' \in G\). In this case, the continuous map, which satisfies \(\pi(gg')=\pi_1(g) \pi_0(g')\) for \(g,g' \in G\) is denoted by \(\pi : G \to B(X_0,X_2)\).
A pair \((G,\ell)\) of a locally compact group with a length function satisfies the property \((*)\) if there exists \(s,t,C >0\) and a sequence \(m_n\) of positive probability measures whose support is contained in \(\{g|\ell(g) \leq n\}\) such that the following holds: Let \((X_0,X_1,X_2,\pi_0,\pi_1)\) be a two-step representation and \(L\) a real number such that \(X_1\) is a Hilbert space and \(\|\pi_i(g)\|\leq L e^{s \ell(g)}\) for all \(g \in G\) and \(i \in \{0,1\}\). Then there is \(P \in B(X_0,X_2)\) such that \(\|\pi(m_n) - P\| \leq C L^2 e^{-tn}\), and such that \(\lim_n \|\pi(\delta_g \ast m_n \ast \delta_{g'}) - \pi(m_n)\|=0\) for every \(g,g'\in G\).
The main result of the paper states that every higher-rank group or lattice in it satisfies property \((*)\). He investigates non-co-compact lattices such as \(SL_n(\mathbb{Z})\) for \(n\geq 3\).
If \(\mathcal{E}\) is a class of Banach spaces it is said that \(G\) satisfies \((*_{\mathcal{E}})\) if in \((*)\) the assumption that \(X_1\) is a Hilbert space is replaced by \(X_1 \in \mathcal{E}\). Another significant result of the paper, which is an extension of some results of V. Lafforgue [J. Topol. Anal. 1, No. 3, 191–206 (2009; Zbl 1186.46022)] and B. Liao [J. Topol. Anal. 6, No. 1, 75–105 (2014; Zbl 1291.22010)], states that if \(G\) is a higher rank simple group over a non-Archimedean local field, or a lattice therein, then \(G\) satisfies \((*_{\mathcal E})\) for every class of Banach spaces \(\mathcal{E}\) of nontrivial type.

MSC:

22D12 Other representations of locally compact groups
22E40 Discrete subgroups of Lie groups
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