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Benjamini-Schramm convergence and zeta functions. (English) Zbl 1470.11135

In a recent paper, D. Lenz et al. proved that a sequence of graphs is Benjamini-Schramm convergent to an infinite tree if, and only if, some zeta function converges to the trivial one [Trans. Am. Math. Soc. 371, No. 8, 5687–5729 (2019; Zbl 1409.05075)]. The paper under review studies this equivalence for arbitrary locally homogeneous spaces.
More precisely, let \(G\) be a locally compact group, \(\{\Gamma_n\}\) a sequence of lattices in \(G\) and \(\Gamma_\infty\) a common normal subgroup. Write \(P_n\) for the normalized Haar measure on \(\Gamma_n\backslash G\). The sequence \(\{\Gamma_n\}\) is said to Benjamini-Schramm converge to \(\Gamma_\infty\) if for every compact set \(C\subset G\) \[\lim_{n\rightarrow\infty}P_n(\{x\in\Gamma_n\backslash G\;:\; x^{-1}(\Gamma_n\backslash\Gamma_\infty)x\cap C=\emptyset\})=1.\] One of the main results in the paper under review can be stated as follows (Proposition 1.3). Suppose \(K\) is a compact subgroup of \(G\) and that \(G/K\) is generated by a \(G\)-invariant proper metric. Consider the following statements:
(a)
The sequence \(\{\Gamma_n\}\) is BS-convergent to \(\Gamma_\infty\).
(b)
The sequence of metric probability spaces \(\{\Gamma_n\backslash G/K\}\) is BS-convergent to \(\Gamma_\infty\backslash G/K\).
Then (a) implies (b), and, under some extra conditions, (b) implies (a).
This extends a previous result of the author in [Enseign. Math. (2) 64, No. 3–4, 371–394 (2018; Zbl 1425.81044)].
Suppose now that \(G\) is a reductive Lie group and \(\Gamma\subset G\) is a cocompact lattice. The unitary representation of \(G\) given by right-translations on \(L^2(\Gamma\backslash G)\) decomposes into a (countable) direct sum of irreducible unitary representations of \(G\) \[L^2(\Gamma\backslash G)=\bigoplus_{\pi\in\widehat{G}}N_{\Gamma}(\pi)\pi\] with finite multiplicities \(N_{\Gamma}(\pi)\). For \(f\) a smooth compactly supported function on \(G\), the operator \[\pi(f):V_\pi\ni v\mapsto\int_Gf(g)\pi(g)vdg\] is of trace class. Then a sequence \(\{\Gamma_n\}\) of torsion-free cocompact lattice in \(G\) is said to be a Plancherel sequence if for every smooth compactly supported function \(f\) on \(G\) one has \[\lim_{n\rightarrow\infty}\frac{1}{\text{vol}(\Gamma_n\backslash G)}\int_{\widehat{G}}\text{tr }\pi(f) d\mu_{\Gamma_n}(\pi)=f(e)\] where \(\mu_{\Gamma_n}:=\sum_{\pi\in\widehat{G}}N_\Gamma(\pi)\delta_\pi\) is the spectral measure attached to \(\Gamma_n\). On the other hand, suppose that \(G=\mathrm{SL}_2({\mathbb R})\). For \(\gamma\in\Gamma_n\) the length of \(\gamma\) is defined by \(l(\gamma)=\text{inf}_{x\in G/K}d(\gamma x,x)\). Observing that \(l(\gamma)>0\), the Selberg zeta function for \(\Gamma_n\) is defined for \(s\in{\mathbb C}\) with \(\text{Re}(s)>1\) as \[Z_n(s)=\Pi_\gamma\Pi_{k\geq 0}(1-e^{-(s+k)l(\gamma)})\] which extends holomorphically to all of \({\mathbb C}\). Here \(\gamma\) runs over all primitive conjugacy classes in \(\Gamma_n\). Now the second main result proved in the paper under review can be stated as follows (Theorem 3.2). If the sequence \(\{\Gamma_n\}\) is uniformly discrete and Plancherel, then \(\lim_{n\rightarrow\infty}\frac{1}{\text{vol}(\Gamma_n\backslash G)}\frac{Z_n^\prime}{Z_n}(s)=0\) when \(\text{Re}(s)>1\). Conversely, if \(\lim_{n\rightarrow\infty}\frac{1}{\text{vol}(\Gamma_n\backslash G)}\frac{Z_n^\prime}{Z_n}(s)=0\) when \(\text{Re}(s)>1\), then \(\{\Gamma_n\}\) is Plancherel.
In the last section, the author suggests several open questions and possible extensions of his results.
Reviewer: Salah Mehdi (Metz)

MSC:

11F72 Spectral theory; trace formulas (e.g., that of Selberg)
22E40 Discrete subgroups of Lie groups
11M36 Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. (explicit formulas)
11M41 Other Dirichlet series and zeta functions
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References:

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