The author studies the homotopy type of the reductive Borel-Serre compactification \(\Gamma \backslash \overline{X}^{RBS}\) of a locally symmetric space \(\Gamma \backslash X\).
In particular, they identify the exit path \(\infty\)-category of \(\Gamma \backslash \overline{X}^{RBS}\) as the nerve of a 1-category, defined entirely in terms of rational parabolic subgroups and unipotent radicals.
Suppose \(\mathbf{G}\) is a connected reductive linear algebraic group defined over \(\mathbb{Q}\) with anisotropic centre over \(\mathbb{Q}\). Let \(\Gamma \subset \mathbf{G}(\mathbb{Q})\) be a neat arithmetic subgroup and let \(X\) be the associated symmetric space. Assume \(\mathbf{G}\) has positive \(\mathbb{Q}\)-rank (i.e. \(\Gamma \backslash X\) is non-compact). Then their main result (Theorem 4.3) is that
“The exit path \(\infty\)-category of the reductive Borel-Serre compactification \(\Gamma \backslash \overline{X}^{RBS}\) is equivalent to the nerve of its homotopy category. This is in turn is equivalent to the category \(\mathscr{C}_{\Gamma}^{RBS}\) with objects the rational parabolic subgroups of \(\textbf{G}\) and hom-sets \[ \mathscr{C}_{\Gamma}^{RBS}(\mathbf{P}, \mathbf{Q}) = \{ \gamma \in \Gamma \mid \gamma \mathbf{P} \gamma^{-1} \leq \mathbf{Q} \} / \Gamma_{\mathbf{N_p}} \] for all \(\mathbf{P}\), \(\mathbf{Q} \in \mathscr{P}\), where \(\Gamma_{\mathbf{N_P}}\) acts by right multiplication and composition is given by multiplication of representatives.”
In the above, \(\mathscr{P}\) is the poset of rational parabolics of \(\mathbf{G}\), \(\mathbf{N_P}\) is the unipotent radical of the parabolic \(\mathbf{P}\) and \(\Gamma_{\mathbf{N_P}}\) denotes \(\Gamma \cap \mathbf{N_P}(\mathbb{Q})\).
As a consequence (Corollary 5.2), they are able to show that the fundamental group of \(\Gamma \backslash \overline{X}^{RBS}\) is given by \(\Gamma / E_{\Gamma}\), where \(E_{\Gamma}\) is generated by all the rational parabolic subgroups \(\Gamma_{\mathbf{N_P}}\), recovering a result of L. Ji et al. [Asian J. Math. 19, No. 3, 465–486 (2015; Zbl 1329.20063)].
The also give a combinatorial description of constructible complexes of sheaves of \(\Gamma \backslash \overline{X}^{RBS}\) in terms of elements in a category of derived functors (Corollary 5.6).
Many of the calculations use tools developed out of the ideas of J. Woolf [J. Homotopy Relat. Struct. 4, No. 1, 359–387 (2009; Zbl 1204.18008)] (in particular, identifying mapping spaces in the exit path \(\infty\)-category as fibres of certain fibrations) and the homotopy links arising in Quinn’s theory of homotopically stratified sets [F. Quinn, J. Am. Math. Soc. 1, No. 2, 441–499 (1988; Zbl 0655.57010)].