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**On the growth of \(L^2\)-invariants for sequences of lattices in Lie groups.**
*(English)*
Zbl 1379.22006

This paper concerns the asymptotic behavior of Betti numbers (and some other invariants) for sequences of compact spaces with a common universal cover, with the goal of proving convergence towards \(L^2\)-invariants that are defined via the universal covering space. Such convergence questions had so far mainly been treated for spaces within one commensurability class. In particular there is the approximation theorem of W. Lück [Geom. Funct. Anal. 4, No. 4, 455–481 (1994; Zbl 0853.57021)], which asserts that \(\lim_{n\to\infty}\frac{b_k(M_n)}{d_n}=b_k^{(2)}(M)\) for a sequence of coverings \(M_n\to M\) of finite degree \(d_n\), with \(d_n\) going to infinity. The results of the paper under review are in many cases getting rid of the restriction on considering only finite coverings of one fixed manifold.

The general setting, in which the authors prove their results, is the notion of Benjamini-Schramm convergence. Let \({\mathcal M}\) be the space of pointed, proper metric spaces with the pointed Gromov-Hausdorff topology. A Riemannian manifold \(M\) yields a measure \(\mu_M\) on \({\mathcal M}\) by considering all pointed spaces with underlying space \(M\) and distributing the pointed spaces along the Riemannian measure on \(M\), i.e., considering the push-forward of the Riemannian measure under the natural map \(M\to{\mathcal M}\) sending a point to the pointed space. A sequence of manifolds \(M_n\) is said to BS-converge if the sequence \(\mu_{M_n}\) converges weakly.

Let now \(M_n=\Gamma_n\backslash X\) be a sequence of locally symmetric spaces modeled after the symmetric space of noncompact type \(X=G/K\). One of the main results of the paper is the proof of the equality \(\lim_{n\to\infty}\frac{b_k(M_n)}{\mathrm{vol}(M_n)}=\beta_k^{(2)}(X)\) under the two assumptions that the sequence \(M_n\) BS-converges to \(X\) and that the sequence \(\Gamma_n\) is uniformly discrete, i.e., the injectivity radii of the \(M_n\) are uniformly bounded away from zero.

In their earlier announcement [C. R., Math., Acad. Sci. Paris 349, No. 15–16, 831–835 (2011; Zbl 1223.53039)] the authors had proven this result by using that Betti numbers resp. \(L^2\)-Betti numbers can be computed via traces of heat kernels resp. \(L^2\)-heat kernels, and by proving an inequality \(| \mathrm{tr} e^{-t\Delta_k}(x,x)-\mathrm{tr} e^{-t\Delta_k^{(2)}}(x,x)| \leq c\cdot \mathrm{InjRad}_M(x)^{-d}\).

The paper under review takes a new approach in that it considers the relative Plancherel measure on \(\widehat{G}\), which counts occurrences of representations of \(G\) in the right regular representation \(L^2(\Gamma_n\backslash G)\). The result proven is that under the above assumptions on BS-convergence and uniform discreteness these relative Plancherel measures converge (after normalizing them by dividing through \(\mathrm{vol}(\Gamma_n\backslash G)\)) to the absolute Plancherel measure on \(\widehat{G}\), which counts occurrences of representations of \(G\) in the right regular representation \(L^2(G)\).

BS-convergence of \(M_n=\Gamma_n\backslash X\) to \(X\) is equivalent to demanding that for any \(R>0\) one has \(\lim_{n\to\infty}\frac{\mathrm{vol}((M_n)_{\leq R})}{\mathrm{vol}(M_n)}=0\), where \((M_n)_{\leq R}\) is the \(R\)-thin part of \(M_n\), and it is also equivalent to demanding that the invariant random subgroups \(\mu_{\Gamma_n}\) converge to \(\mu_{id}\). Here, an invariant random subgroup (IRS) is a conjugation-invariant probability measure on the space of subgroups of \(G\), and \(\mu_{\Gamma_n}\) means the conjugation-invariant measure supported on the conjugacy class of \(\Gamma_n\). The authors use this algebraic formulation to prove BS-convergence of the sequence \(M_n=\Gamma_n\backslash X\) towards \(X=G/K\) if \(G\) has property (T) and real rank at least two. Namely, they prove that in this case the ergodic IRSs are exactly \(\mu_{id}, \mu_G\) and \(\mu_\Gamma\) for lattices \(\Gamma\subset G\), and that the set of ergodic IRSs is compact and its only accumulation point is \(\mu_{id}\).

The latter result does not hold if \(X\) were a rank one symmetric space. On the other hand, the authors take up their approach via heat kernels to show that for sequences of hyperbolic \(d\)-manifolds one can get rid of the condition on injectivity radii (albeit not on the condition on BS-convergence) to prove \(\lim_{n\to\infty}\frac{b_k(M_n)}{\mathrm{vol}(M_n)}=\beta_k^{(2)}({\mathbb H}^d)\).

For a compact quotient of \(\mathrm{SL}(m,{\mathbb R})/\mathrm{SO}(m), m\geq 3\) and a sequence of distinct finite covers \(M_n\), the main theorem implies the new result \(\lim_{n\to\infty}\frac{b_k(M_n)}{\mathrm{vol}(M_n)}=0\). When restricting to congruence covers \(M_n\) of a fixed arithmetic \(d\)-manifold \(M\), the authors can show a stronger quantitative version of this result: for \(| k-\frac{d}{2}|>rk_{\mathbb C}(G)-rk_{\mathbb C}(K)\) they obtain the inequality \(b_k(M_n)< C\cdot \mathrm{vol}(M_n)^{1-\alpha}\) for certain constants \(C,\alpha>0\).

Conjecturally the behavior of Reidemeister or analytic torsion \(T_{M_n}\) under BS-convergence to \(X\) (assuming again uniform discreteness) should be similar to the behavior of Betti numbers: for a fixed finite-dimensional representation \(\rho\) of \(G_{\mathbb C}\) the sequence \(\frac{\log T_{M_n}(\rho)}{\mathrm{vol}(M_n)}\) should converge to the \(L^2\)-torsion \(t_X^{(2)}(\rho)\). For strongly acyclical representations this conjecture was proven by N. Bergeron and A. Venkatesh [J. Inst. Math. Jussieu 12, No. 2, 391–447 (2013; Zbl 1266.22013)]. (That paper actually considered only normal coverings, but the proof adapts to the setting of Benjamini-Schramm convergence.)

The analytical torsion of the trivial representation is known to be related to torsion in homology, in particular for hyperbolic \(3\)-manifolds one can reformulate the conjecture into the question whether \(\frac{\log| H_1(M_n,{\mathbb Z})_{\mathrm{tors}}|}{\mathrm{vol}(M_n)}\) converges to the \(L^2\)-torsion of hyperbolic \(3\)-space. This shows that the assumption on uniform discreteness can not be neglected because J. F. Brock and N. M. Dunfield [Geom. Topol. 19, No. 1, 497–523 (2015; Zbl 1312.57022)] have constructed a sequence of (not uniformly discrete) hyperbolic \(3\)-manifolds, which are integer homology spheres and which BS-converge towards hyperbolic \(3\)-space.

The general setting, in which the authors prove their results, is the notion of Benjamini-Schramm convergence. Let \({\mathcal M}\) be the space of pointed, proper metric spaces with the pointed Gromov-Hausdorff topology. A Riemannian manifold \(M\) yields a measure \(\mu_M\) on \({\mathcal M}\) by considering all pointed spaces with underlying space \(M\) and distributing the pointed spaces along the Riemannian measure on \(M\), i.e., considering the push-forward of the Riemannian measure under the natural map \(M\to{\mathcal M}\) sending a point to the pointed space. A sequence of manifolds \(M_n\) is said to BS-converge if the sequence \(\mu_{M_n}\) converges weakly.

Let now \(M_n=\Gamma_n\backslash X\) be a sequence of locally symmetric spaces modeled after the symmetric space of noncompact type \(X=G/K\). One of the main results of the paper is the proof of the equality \(\lim_{n\to\infty}\frac{b_k(M_n)}{\mathrm{vol}(M_n)}=\beta_k^{(2)}(X)\) under the two assumptions that the sequence \(M_n\) BS-converges to \(X\) and that the sequence \(\Gamma_n\) is uniformly discrete, i.e., the injectivity radii of the \(M_n\) are uniformly bounded away from zero.

In their earlier announcement [C. R., Math., Acad. Sci. Paris 349, No. 15–16, 831–835 (2011; Zbl 1223.53039)] the authors had proven this result by using that Betti numbers resp. \(L^2\)-Betti numbers can be computed via traces of heat kernels resp. \(L^2\)-heat kernels, and by proving an inequality \(| \mathrm{tr} e^{-t\Delta_k}(x,x)-\mathrm{tr} e^{-t\Delta_k^{(2)}}(x,x)| \leq c\cdot \mathrm{InjRad}_M(x)^{-d}\).

The paper under review takes a new approach in that it considers the relative Plancherel measure on \(\widehat{G}\), which counts occurrences of representations of \(G\) in the right regular representation \(L^2(\Gamma_n\backslash G)\). The result proven is that under the above assumptions on BS-convergence and uniform discreteness these relative Plancherel measures converge (after normalizing them by dividing through \(\mathrm{vol}(\Gamma_n\backslash G)\)) to the absolute Plancherel measure on \(\widehat{G}\), which counts occurrences of representations of \(G\) in the right regular representation \(L^2(G)\).

BS-convergence of \(M_n=\Gamma_n\backslash X\) to \(X\) is equivalent to demanding that for any \(R>0\) one has \(\lim_{n\to\infty}\frac{\mathrm{vol}((M_n)_{\leq R})}{\mathrm{vol}(M_n)}=0\), where \((M_n)_{\leq R}\) is the \(R\)-thin part of \(M_n\), and it is also equivalent to demanding that the invariant random subgroups \(\mu_{\Gamma_n}\) converge to \(\mu_{id}\). Here, an invariant random subgroup (IRS) is a conjugation-invariant probability measure on the space of subgroups of \(G\), and \(\mu_{\Gamma_n}\) means the conjugation-invariant measure supported on the conjugacy class of \(\Gamma_n\). The authors use this algebraic formulation to prove BS-convergence of the sequence \(M_n=\Gamma_n\backslash X\) towards \(X=G/K\) if \(G\) has property (T) and real rank at least two. Namely, they prove that in this case the ergodic IRSs are exactly \(\mu_{id}, \mu_G\) and \(\mu_\Gamma\) for lattices \(\Gamma\subset G\), and that the set of ergodic IRSs is compact and its only accumulation point is \(\mu_{id}\).

The latter result does not hold if \(X\) were a rank one symmetric space. On the other hand, the authors take up their approach via heat kernels to show that for sequences of hyperbolic \(d\)-manifolds one can get rid of the condition on injectivity radii (albeit not on the condition on BS-convergence) to prove \(\lim_{n\to\infty}\frac{b_k(M_n)}{\mathrm{vol}(M_n)}=\beta_k^{(2)}({\mathbb H}^d)\).

For a compact quotient of \(\mathrm{SL}(m,{\mathbb R})/\mathrm{SO}(m), m\geq 3\) and a sequence of distinct finite covers \(M_n\), the main theorem implies the new result \(\lim_{n\to\infty}\frac{b_k(M_n)}{\mathrm{vol}(M_n)}=0\). When restricting to congruence covers \(M_n\) of a fixed arithmetic \(d\)-manifold \(M\), the authors can show a stronger quantitative version of this result: for \(| k-\frac{d}{2}|>rk_{\mathbb C}(G)-rk_{\mathbb C}(K)\) they obtain the inequality \(b_k(M_n)< C\cdot \mathrm{vol}(M_n)^{1-\alpha}\) for certain constants \(C,\alpha>0\).

Conjecturally the behavior of Reidemeister or analytic torsion \(T_{M_n}\) under BS-convergence to \(X\) (assuming again uniform discreteness) should be similar to the behavior of Betti numbers: for a fixed finite-dimensional representation \(\rho\) of \(G_{\mathbb C}\) the sequence \(\frac{\log T_{M_n}(\rho)}{\mathrm{vol}(M_n)}\) should converge to the \(L^2\)-torsion \(t_X^{(2)}(\rho)\). For strongly acyclical representations this conjecture was proven by N. Bergeron and A. Venkatesh [J. Inst. Math. Jussieu 12, No. 2, 391–447 (2013; Zbl 1266.22013)]. (That paper actually considered only normal coverings, but the proof adapts to the setting of Benjamini-Schramm convergence.)

The analytical torsion of the trivial representation is known to be related to torsion in homology, in particular for hyperbolic \(3\)-manifolds one can reformulate the conjecture into the question whether \(\frac{\log| H_1(M_n,{\mathbb Z})_{\mathrm{tors}}|}{\mathrm{vol}(M_n)}\) converges to the \(L^2\)-torsion of hyperbolic \(3\)-space. This shows that the assumption on uniform discreteness can not be neglected because J. F. Brock and N. M. Dunfield [Geom. Topol. 19, No. 1, 497–523 (2015; Zbl 1312.57022)] have constructed a sequence of (not uniformly discrete) hyperbolic \(3\)-manifolds, which are integer homology spheres and which BS-converge towards hyperbolic \(3\)-space.

Reviewer: Thilo Kuessner (Augsburg)