Geometric construction of homology classes in Riemannian manifolds covered by products of hyperbolic planes (Q2039260)

This paper is dedicated to the study of the homology of arithmetic lattices in the Lie groups \(\mathrm{SL}_2(\mathbb R)^r\), or equivalently of the associated locally symmetric spaces, which are quotients of a product \((\mathbb H^2)^r\) of hyperbolic planes. It is known by indirect arguments that the middle Betti number \(b_r\) for such spaces can get arbitrarily large, and in fact it is roughly proportional to the volume, at least assuming the congruence property for such lattices. The main result here is a constructive version of the non-quantitative version of this result. The exact statement is the following: if \(M\) is an arithmetic quotient of \((\mathbb H^2)^r\), any maximal compact flat submanifold \(T\) (a \(r\)-dimensional flat torus) can be lifted in some finite cover to any number of linearly independent homology classes. The proof given here is purely geometric, relying on the construction of embedded lifts with a good pattern of intersections, the methods being inspired by those used in [\textit{G. Avramidi} and \textit{T. T. Nguyễn-Phan}, Geom. Funct. Anal. 29, No. 6, 1638--1702 (2019; Zbl 1432.53050)]. To finish this review we note that the main theorem in the particular case of Hilbert modular surfaces (the noncompact arithmetic quotients of \((\mathbb H^2)^r\)) is also a consequence of the results in [\textit{N. Bergeron}, in: Séminaire de théorie spectrale et géométrie. Année 2002--2003. St. Martin d'Hères: Université de Grenoble I, Institut Fourier. 75--101 (2003; Zbl 1053.11047)].