Generalized Cusps in Real Projective Manifolds: Classification

A generalized cusp C is diffeomorphic to [0,∞) times a closed Euclidean manifold. Geometrically C is the quotient of a properly convex domain by a lattice, Γ, in one of a family of affine groups G(ψ), parameterized by a point ψ in the (dual closed) Weyl chamber for SL(n+1,R), and Γ determines the cusp up to equivalence. These affine groups correspond to certain fibered geometries, each of which is a bundle over an open simplex with fiber a horoball in hyperbolic space, and the lattices are classified by certain Bieberbach groups plus some auxiliary data. The cusp has finite Busemann measure if and only if G(ψ) contains unipotent elements. There is a natural underlying Euclidean structure on C unrelated to the Hilbert metric. A generalized cusp is a properly convex projective manifold C = Ω/Γ where Ω ⊂ RP is a properly convex set and Γ ⊂ PGL(n+ 1,R) is a virtually abelian discrete group that preserves Ω. We also require that ∂C is compact and strictly convex (contains no line segment) and that there is a diffeomorphism h : [0,∞)× ∂C −→ C. An example is a cusp in a hyperbolic manifold that is the quotient of a horoball. Another generalized cusp C ′ = Ω′/Γ′ is equivalent to C if there is a generalized cusp C ′′ and projective embeddings, that are also homotopy equivalences, of C ′′ into both C and C ′, and they are all diffeomorphic. It follows from the classification theorem that generalized cusps are equivalent if and only if Γ and Γ′ are conjugate subgroups of PGL(n+ 1,R). It follows from [10] that every generalized cusp in a strictly convex manifold of finite volume is equivalent to a standard cusp, i.e. a cusp in a hyperbolic manifold. A generalized cusp is homogeneous if PGL(Ω) (the group of projective transformations that preserves Ω) acts transitively on ∂Ω. It was shown in [11] that every generalized cusp is equivalent to a homogeneous one and, that if the holonomy of a generalized cusp contains no hyperbolic elements, then it is equivalent to a standard cusp. Furthermore, by [11] it follows that generalized cusps often occur as ends of properly convex manifolds obtained by deforming finite volume hyperbolic manifolds. The holonomy of a generalized cusp is conjugate to a lattice in one of a family of Lie subgroups G(ψ) ⊂ PGL(n + 1,R), parameterized by ψ ∈ Hom(R,R) with ψ(e1) ≥ ψ(e2) ≥ · · ·ψ(en) ≥ 0. Elements of the unipotent subgroup P (ψ) ⊂ G(ψ) are called parabolic. The type t = tψ, is the number of i with ψ(ei) 6= 0, and the unipotent rank is u(ψ) = dimP (ψ) = max(n− t− 1, 0). The group G(ψ) is called a cusp Lie group and preserves a properly-convex domain, Ω(ψ), together with a convex function hψ : Ω(ψ) → R. The level sets Ht = h−1 ψ (t) are G(ψ)-orbits, and are convex hypersurfaces called horospheres. The horospheres with t ≤ 0 foliate Ω(ψ). There is a transverse G(ψ)-foliation by a pencil of lines, and a one parameter subgroup of PGL(n + 1,R), called the radial flow, that preserves each line, normalizes G(ψ), and permutes the horospheres. The interior of Ω(0) is a model of hyperbolic space, H, and ∂Ω(0) = Sn−1 ∞ − z where z is some point in Sn−1 ∞ . Moreover G(0) ⊂ Isom(H) is the group generated by parabolics and elliptics that fix z. At the other extreme, when t = n, then G(ψ) contains a finite index subgroup that is diagonalizable. Moreover, G(ψ) = PGL(Ω(ψ)) if ψ 6= 0. When t < n, the geometry (Ω(ψ), G(ψ)) is fibered over a simplex ∆ with fiber a horoball in H, see (1.32). If Γ ⊂ G(ψ) is a lattice then C = Ω(ψ)/Γ is called a ψ-cusp. This is a projective manifold if Γ is torsion free, and in general is a projective orbifold. In this paper we have chosen to discuss only Date: October 10, 2017. 1 ar X iv :1 71 0. 03 13 2v 1 [ m at h. G T ] 9 O ct 2 01 7 2 SAMUEL A. BALLAS, DARYL COOPER, AND ARIELLE LEITNER manifolds, though everything works (suitably modified) for orbifolds. The image of a horosphere in C is called a horomanifold and these foliate C. Theorem 0.1 (uniformization). Every generalized cusp is equivalent to a ψ-cusp. The next goal is to classify cusps up to equivalence. For this it is useful to introduce marked cusps and marked lattices (see Section 3 for the definition and more discussion). A rank-2 cusp in a hyperbolic 3-manifold is determined by a cusp shape, which is a Euclidean torus defined up to similarity. This shape is usually described by a complex number x + iy with y > 0, that uniquely determines a marked cusp. Unmarked cusps are described by the modular surface H/PSL(2,Z). More generally, a maximal-rank cusp in a hyperbolic n-manifold is determined by a lattice in Isom(En−1) up to conjugacy and rescaling. We extend this result by showing when ψ 6= 0 that a generalized cusp of dimension n with holonomy in G(ψ) is determined by a pair ([Γ], A · O(ψ)) consisting of the conjugacy class of a lattice Γ ⊂ Isom(En−1), and an anisotropy parameter which we now describe. The second fundamental form on ∂Ω is a Euclidean metric that is preserved by the action of G(ψ). This identifies G(ψ) with a subgroup of Isom(En−1), and G(ψ) = T (ψ)oO(ψ) is the semidirect product of the translation subgroup, T (ψ) ∼= Rn−1, and a closed subgroup O(ψ) ⊂ O(n− 1) that fixes some point p in ∂Ω, see (1.22). The Euclidean structure identifies Γ with a lattice in Isom(En−1). This lattice is unique up to conjugation by an element of O(ψ). The anisotropy parameter is a left coset A · O(ψ) in O(n) that determines the O(ψ)-conjugacy class. The group O(ψ) is computed in (1.21). Given a Lie group G, the set of G-conjugacy classes of marked lattices in G is denoted T (G). Define T (Isom(En−1), ψ) ⊂ T (Isom(En−1)) to be the subset of conjugacy classes of marked Euclidean lattices with rotational part of the holonomy (up to conjugacy) in O(ψ). The classification of generalized cusps (up to equivalence) is completed by: Theorem 0.2 (classification). (1) If Γ and Γ′ are lattices in G(ψ) TFAE (a) Ω(ψ)/Γ and Ω(ψ)/Γ′ are equivalent generalized cusps (b) Γ and Γ′ are conjugate in PGL(n+ 1,R) (c) Γ and Γ′ are conjugate in PGL(Ω(ψ)) (2) A lattice in G(ψ) is conjugate in PGL(n+ 1,R) into G(ψ′) iff G(ψ) is conjugate to G(ψ′). (3) G(ψ) is conjugate in PGL(n+ 1,R) to G(ψ′) iff ψ′ = t · ψ for some t > 0. (4) PGL(Ω(ψ)) = G(ψ) when ψ 6= 0 (5) When ψ 6= 0 the map Θ : T (Isom(En−1), ψ) × (O(n − 1)/O(ψ)) −→ T (G(ψ)) defined in (29) is a bijection. One might view (2) in the context of super-rigidity : an embedding of a lattice determines an embedding of the Lie group that contains it. Throughout this paper we repeatedly stumble over two exceptional cases. A generalized cusp with ψ = 0 is projectively equivalent to a cusp in a hyperbolic manifold. This is the only case when PGL(Ω(ψ)) is strictly larger than G(ψ), and is caused by elements of PGL(Ω(0)) ⊂ Isom(H) that permute horospheres. These elements are hyperbolic isometries of H that fix z. This accounts for the fact that the equivalence class of a cusp in a hyperbolic manifold is determined by the similarity class (PGL(Ω(ψ))-conjugacy class) of the lattice, rather than the G(ψ)-conjugacy class, as in every other case. The other exceptional case is the diagonalizable case t = n, and in this case the radial flow is hyperbolic instead of parabolic. Fortunately both these exceptional cases are easy to understand, but tend to require proofs that consider various cases. Let Cn denote the set of equivalence classes of generalized cusps of dimension n. Let Mod denote the (disjoint) union over all ψ with ψ(e1) = 1 of conjugacy classes of (unmarked) lattice in G(ψ), union lattices in G(0) ∼= Isom(En−1) up to conjugacy and scaling. It follows from the above that every non-standard generalized cusp is equivalent to one given by a lattice in G(ψ) with ψ(e1) = 1, that is unique up to conjugacy in G(ψ) giving: