Arithmetic cusp shapes are dense

In this article we verify an orbifold version of a conjecture of Nimershiem from 1998. Namely, for every flat n–manifold M, we show that the set of similarity classes of flat metrics on M which occur as a cusp cross-section of a hyperbolic (n+1)–orbifold is dense in the space of similarity classes of flat metrics on M. The set used for density is precisely the set of those classes which arise in arithmetic orbifolds. 1 Main results By a flat n–manifold we mean a closed manifold M = Rn/Γ where Γ is a discrete, torsion free, cocompact subgroup of Isom(Rn). In analogy with Teichmüller theory, there is a contractible space F(M) of flat metrics on M coming from the standard flat structure on Rn and all the possible Isom(Rn)–conjugacy classes for Γ. We say two flat metrics g1,g2 on M are similar if there exists α ∈ R and an isometry between (M,αg1) and (M,g2). We denote the equivalence class under similarity of a flat metric g by [g] and the space of similarity classes of flat metrics on M by S(M). An important relationship between flat and hyperbolic geometry is exhibited in the thick-thin decomposition of a hyperbolic manifold. Specifically, every finite volume, noncompact hyperbolic (n + 1)–orbifold W has a thick-thin decomposition comprised of a compact manifold Wcore with boundary components M1, . . . ,Mm and manifolds E1, . . . ,Em of the form M j ×R≥0. The manifolds E j are called cusp ends, the manifolds M j are called cusp cross-sections, and the union of Wcore along the boundary with E1, . . . ,Em recovers W topologically. The manifolds M j are known to be flat n–manifolds and are totally geodesic boundary components of the manifold Wcore equipped with the quotient metric coming from the path metric on its neutered space N ⊂ Hn+1. Indeed, a cusp cross-section M j is naturally furnished with a flat metric g which is well-defined up to similarity and we call these similarity classes of flat metrics realizable flat similarity classes or cusp shapes. This article is devoted to the classification of the possible cusp shapes of a flat n–manifold occurring in the class of arithmetic (n + 1)–orbifolds. The absence of a general geometric construction for hyperbolic orbifolds forces our restriction to orbifolds produced by arithmetic means. ∗Supported by a V.I.G.R.E graduate fellowship and Continuing Education fellowship. Arithmetic cusp shapes are dense 2 Given this forced restriction, the picture we provide here is complete. Before stating our main results, we briefly survey some preexisting results and questions. Motivated by the above picture and work of Gromov [3], Farrell and Zdravkovska [2] conjectured every flat n–manifold arises as a cusp cross-section of a 1–cusped hyperbolic (n+1)–manifold and this is easily verified for n = 2. Indeed, the complement of a knot in S3 is typically endowed with a finite volume, complete hyperbolic structure with one cusp (see [11]), and thus gives the realization of the 2–torus T 2 as a cusp cross-section of a 1–cusped hyperbolic 3–manifold. Likewise, the Klein bottle arises as a cusp cross-section of the 1–cusped Gieseking manifold (see [10]). However, Long and Reid [5] constructed counterexamples for n = 3 by showing any flat 3–manifold arising as a cusp cross-section of a 1–cusped hyperbolic 4–manifold must have integral η–invariant; this works for all n = 4k−1. The failure of the conjecture of Farrell–Zdravkovska is far from total. Nimershiem [8] showed every flat 3–manifold arises as a cusp cross-section of a hyperbolic 4–manifold, and Long and Reid [6] proved every flat n–manifold arises as a cusp cross-section of an arithmetic hyperbolic (n + 1)–orbifold. A more geometrically relevant question is precisely which cusp shapes occur on a given flat n–manifold. Via a counting argument, almost every similarity class on a flat n–manifold must fail to appear as a cusp shape. Despite this, Nimershiem [8] showed any for flat 3–manifold M the cusp shapes occurring in hyperbolic 4–manifolds are dense in the space of flat similarity classes on M, and conjectured [8, Conj. 2’] this for every flat n–manifold. Our main result is the verification of this conjecture in the orbifold category. Theorem 1.1 (Cusp shape density). For a flat n–manifold M, the set of cusp shapes of M occurring in hyperbolic (n + 1)–orbifolds is dense in the space of flat similarity classes S(M). It is worth mentioning that the proof of Theorem 1.1 exhibits a dense subset of shapes of a uniform nature. These similarity classes are the image of Q–points of a Q–algebraic set under a projection map. From this one sees density occurs not as a function of small complexity in low dimensions but from the algebraic structure of these spaces. Moreover, these similarity classes of flat metrics are precisely those classes which occur in the cusp cross-sections of arithmetic hyperbolic (n + 1)–orbifolds—see Theorem 3.7. Using a modest refinement of Selberg’s lemma, we verify the full conjecture for the n–torus. Theorem 1.2. For the n–torus Rn/Zn, the set of cusp shapes of Rn/Zn occurring in hyperbolic (n+1)–manifolds is dense in the space of flat similarity classes S(Rn/Zn).