Cusps of Hilbert modular varieties

Abstract

Motivated by a question of Hirzebruch on the possible topological types of cusp cross-sections of Hilbert modular varieties, we give a necessary and sufficient condition for a manifold M to be diffeomorphic to a cusp cross-section of a Hilbert modular variety. Specialized to Hilbert modular surfaces, this proves that every Sol 3-manifold is diffeomorphic to a cusp cross-section of a (generalized) Hilbert modular surface. We also deduce an obstruction to geometric bounding in this setting. Consequently, there exist Sol 3-manifolds that cannot arise as a cusp crosssection of a 1-cusped nonsingular Hilbert modular surface.

Cusp shapes of Hilbert–Blumenthal surfaces

“…This admits the topological interpretation that M ∆ is diffeomorphic to T 3 ϕ ˆR`, where T 3 ϕ is the infrasolv manifold (in the sense of [17, §2.4.3]) that fibers over the circle, with fiber the torus, and Anosov diffeomorphism ϕ [18]. We call T 3 ϕ the cusp section of M Γ (or equivalently, of Γ).…”

“…While these advances have been fruitful in understanding arithmetic and topological properties, certain geometric properties had remained elusive. A cusp (cross) section of M Γ is a 3-dimensional mapping torus of some Anosov diffeomorphism ϕ of the torus and, due to McReynolds [17,18], every Sol 3-manifold is commensurable to one of these cusp sections up to diffeomorphism. However, there had previously been no combinatorial description of these in terms of their sides and the action of ϕ as an explicit side-pairing map.…”

Controlling manifold covers of orbifolds

“…For cusp cross sections of higher rank locally symmetric spaces, the fundamental group of a cusp cross section is virtually solvable but typically not virtually unipotent. For instance, cusp cross sections of Hilbert modular surfaces are Sol 3-orbifolds (see [17] for more on this). Though Theorem 1.1 might not hold for these groups, Corollary 1.2 extends.…”