Torsion homology growth in arithmetic groups
This survey is rich with ideas. The author expresses some regret that various questions that could have been included, could not be covered in the article. But, the ideas, motivations to questions, known results, conjectures and approaches towards them that are sketched here are already a veritable treasure for non-experts as well as useful to experts. The results and conjectures described in the article have been obtained in collaboration with Akshay Venkatesh. Classically, only the free part of the homology groups of arithmetic groups was studied in detail. For instance, for congruence subgroups \(\Gamma_0(N)\) of \(\mathrm{SL}_2(R)\) where \(R\) is either \(\mathbb{Z}\) or the ring of integers of a real quadratic field, the free part \(H_i(\Gamma_0(N), \mathbb{Z})\) for \(i=1,2\) is intimately related to classical and Hilbert modular forms respectively. The torsion part appears to be little arithmetic interest. However, in marked contrast, when \(R\) is the ring of integers of an imaginary quadratic field, the free part is small but the torsion part is large and has deep arithmetic interest. To give a few dramatic examples due to \textit{M. H. Şengün} [Int. J. Number Theory 8, No. 2, 311--320 (2012; Zbl 1243.30085)]: When \(R = \mathbb{Z}[i]\), then \[ \Gamma_0(41+56i)^{ab} \cong \mathbb{Z}/4078793513671 \mathbb{Z} \oplus \mathbb{Z}/292306033 \mathbb{Z} \oplus \cdots \] \[ \Gamma_0(118+175i)^{ab} \cong \mathbb{Z} \oplus T \] where \(T\) is a finite abelian group of order \(>10^{310}\). As the author puts it, ``beyond the guilty pleasure of exhibiting gigantic randomly distributed primes, there are more serious reasons to study torsion in the homology''. Concerning congruence subgroups of \(\mathrm{SL}_2(R)\) for \(R\) ring of integers of an imaginary quadratic field, the following conjecture is stated: \textbf{Conjecture.} Let \(M_n \rightarrow M_0\) (\(n \in N\)) be a sequence of congruence covers of a fixed congruence hyperbolic \(3\)-manifold \(M_0\) such that \(\mathrm{Vol}(M_n) \rightarrow \infty\). Then, \[ \lim_{n \rightarrow \infty} \frac{\log|H_1(M_n, \mathbb{Z})_{\mathrm{tors}}|}{\mathrm{Vol}(M_n)} = \frac{1}{6 \pi}. \] A particular case of this conjecture for \(\mathrm{SL}_2(\mathbb{Z}[i])\) asserts more precisely that: \[ \frac{\log |H_1(\Gamma_0(N),\mathbb{Z})_{\mathrm{tors}}|}{N^2} \rightarrow \frac{\lambda}{18 \pi} \] as \(N \rightarrow \infty\), where \(\lambda = L(\chi_{\mathbb{Q}(i)},2) = 1 - \frac{1}{9} + \frac{1}{25} - \cdots\) The conjecture motivated the computations of Sengün mentioned above. The motivation for the conjecture itself comes from the study of Ray-Singer analytic torsion of a closed manifold, where a special case of a theorem of Cheeger and Müller implies an expression in terms of the Selberg zeta function that is analogous to expressing the central value of the $L$-function of an elliptic curve in terms of the rank and size of the Tate-Shafarevich group. The author sketches an argument to explain that mod-\(p\) torsion classes in \(\Gamma_0(N)\) (very roughly) parametrize quadratic extensions of \(\mathbb{Q}(\sqrt{d})\) whose Galois group is a subgroup of \(\mathrm{GL}_2(\overline{\mathbb{F}_p})\). Conversely, certain conjectures that associate torsion classes to some field extensions are analyzed with a view to explaining the drastically different behaviour exhibited in the real quadratic and the imaginary quadratic cases. Motivated by this, the author considers posing a suitable conjecture for general arithmetic groups and states the following precise version for the case of subgroups of finite index in \(\mathrm{SL}_n(\mathbb{Z})\) for \(n \geq 3\). \textbf{Conjecture.} Let \(\Gamma_i\) (\(i \geq 1\)) be a family of distinct subgroups of finite index in \(\mathrm{SL}_n(\mathbb{Z})\). Then, as \(i \rightarrow \infty\), the numbers \(\frac{\log|H_q(\Gamma_i, \mathbb{Z})_{tors}|}{[\mathrm{SL}_n(\mathbb{Z}):\Gamma_i]}\) tend to \(0\) unless we are in one of the following two situations: \(n=3, q=2\) in which case the limit is \(\frac{\zeta(3)}{288 \pi^2}\); or \(n=4=q\) i which case the limit is \(\frac{31 \sqrt{2} \zeta(3)}{259200 \pi^2}\). The author also goes on to sketch a possible method of proving the conjectures. For the entire collection see [Zbl 1396.00017].
scientific article; zbMATH DE number 6979389
scientific article