Sequences of high rank lattices with large systole containing a fixed genus surface group (Q1717439)

Let \(\Gamma\) be a lattice in \(\mathrm{SL}(3,\mathbb{R})\) and consider the locally symmetric space \(\Gamma\backslash\mathrm{SL}(3,\mathbb{R)} /\mathrm{SO}(3)\). The systole (the length of the shortest closed geodesic) of this space will be denoted by \(\mathrm{sys}(\Gamma)\). Based on their earlier work (see [\textit{D. D. Long} and \textit{A. W. Reid}, Ill. J. Math. 60, No. 1, 39--53 (2016; Zbl 1422.22014)] and [\textit{D. D. Long} and \textit{A. W. Reid}, Math. Sci. Res. Inst. Publ. 61, 151--166 (2014; Zbl 1342.57002)]) the authors prove the following theorems. (Theorem 1.1): If \(\Lambda\) is a non-uniform lattice in \(\mathrm{SL}(2,\mathbb{R})\) which is not commensurable with \(\mathrm{SL} (2,\mathbb{Z})\), then \(\Lambda\) is commensurable with a sequence of torsion-free lattices \(\Gamma_{j}\) with \(\mathrm{sys}(\Gamma_{j} )\rightarrow\infty\) where each \(\Gamma_{j}\) contains a thin surface subgroup of fixed genus. (Theorem 1.2): A similar sequence of torsion-free lattices can be constructed for infinitely many incommensurable uniform lattices \(\Lambda\) in \(\mathrm{SL}(2,\mathbb{R})\). The idea of the construction is the following. Let \(\mathcal{O}_{d}\) be the ring of integers of \(\mathbb{Q}(\sqrt{d})\) where \(d\) is a square-free positive integer. Then it is shown that there exist infinitely many primes \(p\) (depending on \(d\)) such that the principal congruence subgroups \(\Lambda_{d}(p):=\ker(\mathrm{SL}(3,\mathcal{O} _{d})\rightarrow\mathrm{SL}(3,\mathcal{O}_{d}/p\mathcal{O}_{d}))\) are torsion-free and contain a surface subgroup of genus \(3\).