A substitute for Kazhdan's property (T) for universal nonlattices (Q6612319)

Let \(R\) be a commutative finitely generated ring. If \(R\) has a unit, then \(SL_n(R)\) with \(n\ge 3\) is a typical example of a group having the Kazhdan's property (T). Moreover, the universal lattice \(EL_n(R)\) possesses it as well if \(n\ge 3\); this lattice is defined as the subgroup in \(SL_n(R)\) generated by the elementary matrices \(E_{ij}(r)\), \(1\le i,j\le n\), \(r\in R\) (this matrix has \(r\) in the \((i,j)\) coordinate and 0 elsewhere).\N\NIf \(R\) is not unital, situation changes. Let \(R = \mathbb{Z}\langle t_1,\dots, t_d\rangle\) be the space of polynomials in \(d\) variables without free term, then \(EL_n(R)\) has no Kazhdan's property.\N\NThe aim of the paper is to prove a weaker property of this group which is however similar to the property (T). To state it, let us recall first the definition of the property (T) itself. Let \(\Gamma\) be a discrete group with a finite set of generators \(S\). Then \(\Gamma\) has property (T) if for every unitary representation \(\pi\) on a Hilbert space \(H\) the existence of \(v\in H\) and \(\delta>0\) such that \(\| \pi(s)v-v\|<\delta \|v\|\), \(s\in S\), implies that \(\pi\) has a nonzero invariant vector.\N\NNow, Corollary A of the article states that for \(n\) large enough, there exists \(\delta>0\) such that every orthogonal representation of \(EL_n(R)\) on a Hilbert space \(H\) having a vector \(v\in H\) with the property \(\|v - \pi( E_{ij}(t_k)) v\| < \delta\|v\|\) has also a ``more invariant'' nonzero vector \(w\), that is, such that \N\[\N\lim_{l\to\infty} \max_{i,j,k} \| w-\pi(E_{ij}(t_k)) \|=0. \N\]\NThe methods develop earlier results of the author characterizing property (T) in terms of sums of squares. In the real group algebra \(\mathbb{R}[\Gamma]\), positive elements are described as sums of squares: \N\[\N\Sigma^2\mathbb{R}[\Gamma] = \{ \sum \xi_i^* \xi_i: \xi_i\in\mathbb{R}[\Gamma] \}.\N\]\NThe combinatorial Laplcaian is defined as \N\[\N\Delta = \frac12 \sum_{s\in S} (1-s)^*(1-s).\N\]\NAs proved by \textit{N. Ozawa} [J. Inst. Math. Jussieu 15, No. 1, 85--90 (2016; Zbl 1336.22008)], \(\Gamma\) has property (T) if and only if there is \(\varepsilon >0\) such that \(\Delta^2 - \varepsilon \Delta\) is positive in \(\mathbb{R}[\Gamma]\).\N\NThis explains the statement of the main theorem: let \(R = \mathbb{Z}\langle t_1,\dots, t_d\rangle\) be defined as above and \(\Gamma=EL_n(R)\). There exist \(n_0\in\mathbb N\) and \(\varepsilon>0\) such that for every \(n\ge n_0\) the combinatorial Laplacians \N\[\N\Delta = \sum_{i\ne j} \sum_{k=1}^d (1-E_{ij}(t_k))^*(1-E_{ij}(t_k)),\N\]\N\[\N\Delta^{(2)} = \sum_{i\ne j} \sum_{k,l=1}^d (1-E_{ij}(t_k t_l))^*(1-E_{ij}(t_k t_l))\N\]\Nsatisfy \N\[\N\Delta^2 - n \varepsilon \Delta^{(2)} \in \overline{ \Sigma^2\mathbb{R}[\Gamma]}, \N\]\Nthe line denoting the closure in the full group \(C^*\)-algebra: \(\xi \in \mathbb{R}[\Gamma]\) is in \(\overline{ \Sigma^2\mathbb{R}[\Gamma]}\) iff \(\xi\ge0\) in \(C^*(\Gamma)\).