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TESSELLATIONS of SOLVMANIFOLDS 1. Introduction TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 350, Number 9, September 1998, Pages 3767{3796 S 0002-9947(98)01980-1 TESSELLATIONS OF SOLVMANIFOLDS DAVE WITTE Abstract. Let A be a closed subgroup of a connected, solvable Lie group G, such that the homogeneous space A G is simply connected. As a special case n of a theorem of C. T. C. Wall, it is known that every tessellation A G=Γof n AGis finitely covered by a compact homogeneous space G0=Γ0.Weprovethat then covering map can be taken to be very well behaved — a “crossed” affine map. This establishes a connection between the geometry of the tessellation and the geometry of the homogeneous space. In particular, we see that every geometrically-defined flow on A G=Γ that has a dense orbit is covered by a n natural flow on G0=Γ0. 1. Introduction Let G be a transitive group of diffeomorphisms of a manifold M.(Forour n purposes, the most important case is when M = R .) The choice of a particular group G endows M with the structure of a homogeneous space. (In other words, the choice of a particular group G represents the choice of a particular geometric structure on M, in the sense of Klein’s Erlangen Program.) We always assume that G is a finite-dimensional Lie group, and that G is connected (or, at the worst, that G has only finitely many connected components). It is well known that M is G-equivariantly diffeomorphic to a coset space A G (or to G/A, if one prefers to have G act on the left) [V, Lem. 2.9.2, p. 76]. \ In this paper, G is usually solvable, in which case the pair (M,G)issaidtobea solvmanifold. More simply, we usually refer to M as a solvmanifold,withthechoice of a particular solvable group G being understood. Now assume M = Rn. If Γ is a properly discontinuous subgroup of G,such that Rn/Γ is compact, then we may refer to Rn/Γasatessellation, because the Γ-translates of a fundamental domain tessellate Rn.IfGis solvable, then Rn/Γis said to be a solvtessellation. Note that a tessellation Rn/Γ is a manifold if Γ acts freely on Rn. Because Γ has a torsion-free subgroup of finite index [R, Lem. 4.6, p. 57], it follows that every solvtessellation has a finite branched cover that is a manifold. (Hence, every tessellation is an “orbifold” [M, Appendix A].) One may study tessellations of solvmanifolds other than Rn, but the most im- portant case is where the solvmanifold is simply connected. Then, because every simply connected solvmanifold is diffeomorphic to Rn (cf. [M1, Prop. 11.2]), there is no real loss in restricting attention to Rn. Received by the editors October 6, 1994 and, in revised form, November 5, 1996. 1991 Mathematics Subject Classification. Primary 22E25, 22E40, 53C30; Secondary 05B45, 20G20, 20H15, 57S20, 57S30. c 1998 American Mathematical Society 3767 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3768 DAVE WITTE The homeomorphism types of solvtessellations are understood (modulo finite covers), because of fundamental work of C. T. C. Wall [Wa, 15B] on aspherical manifolds with polycyclic fundamental group. Namely, Wall’s§ work implies that some finite (branched) cover of Rn/Γ is homeomorphic to a homogeneous space (more precisely, a solvmanifold). (The lecture notes of Farrell and Jones [FJ] provide a survey of related work in surgery theory.) In other words, there is a n finite group F of diffeomorphisms of some solvmanifold G0/Γ0, such that R /Γ n ≈ (G0/Γ0)/F .IfFacts freely, this means that R /Γ is an “infrasolvmanifold.” Theorem 1.1 (Wall, cf. [Wa, 15B] or [FJ, Theorem 2.16]). Let Γ be a properly discontinuous subgroup of a connected,§ solvable Lie group G that acts transitively n n on R .IfR/Γis compact, then there is some discrete subgroup Γ0 of some simply n connected, solvable Lie group G0, such that some finite branched cover of R /Γ is homeomorphic to G0/Γ0. Another way of stating the theorem is that, to construct a representative of every homeomorphism class of solvtessellations of Rn, modulo finite covers, there is no need to consider groups G of any dimension other than n. Furthermore, the representation of G as a group of diffeomorphisms of Rn can be taken to be the action of G on itself by right multiplication. The assumption that G is solvable cannot be eliminated from the theorem. For example, to construct a compact n-manifold of constant negative curvature, one takes G to be the group of orientation-preserving isometries of hyperbolic n- n space H (that is, G ∼= SO(1,n)). The dimension of G is then n(n +1)/2, which is much larger than n. Because Wall’s proof is topological, it does not seem to show any connection n between the geometry of R /Γ and that of G0/Γ0. (Indeed, Wall’s statement of the theorem applies in a much more general setting, where the manifolds involved have no a priori geometric structure.) In the present paper, we give an algebraic proof of the theorem, thus exhibiting the precise relationship between the geometry of the two spaces. n Ideally, one might hope that some covering map π from G0/Γ0 to R /Γwould respect the geometry, in the sense that every local isometry of G0/Γ0 would push n n down, via π, to a local isometry of R /Γ. Since the geometry of R /Γ is represented by the group G, and the geometry of G0/Γ0 is represented by the group G0,this can be restated more precisely by saying that, ideally, there would be a group homomorphism φ: G0 G, such that → (xg)π~ = xπ~ gφ, · n for every x and g in G0,where˜πis the lift of π to a map G0 R of the universal covers. This is equivalent to saying that, ideally, π wouldbeanaffinemap.→ n n Definition 1.2. For any choice of a basepoint x0 R , the action of G on R n ∈ provides a continuous map G R : g x0g. Then, if φ: G0 G is any group homomorphism,wehaveacompositemap→ 7→ → φ g x0g n φ0:G0 G 7→ R , −−−−→ −−−−→ φ which is an affine map. If, in addition, (Γ0) Γ, then φ0 induces a well-defined n ⊂ map φ00 : G0/Γ0 R /Γ, and φ00 is also called an affine map. → Unfortunately, it is not always possible to arrange for the covering map G0/Γ0 n → R /Γ to be affine. (For example, Figure 4.1 shows a solvtessellation that is homeo- License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use TESSELLATIONS OF SOLVMANIFOLDS 3769 morphic to the ordinary 2-torus T2, but does not have the geometric structure of any solvmanifold.) Thus, in some cases, the covering map must distort the geom- etry. It turns out that the obstructions arise from certain “rotations” associated to G, that is, from certain elements in compact, abelian groups of automorphisms of G. To eliminate the obstructions, we add on more rotations to G, in order to cancel the ones that are causing the difficulty. Thus, instead of a homomorphism from G0 into G, we deal with a homomorphism from G0 into the semidirect product T n G, for some compact, abelian subgroup T of Aut G. This leads to “crossed homomorphisms” and the corresponding “crossed affine maps.” Definition 1.3 (cf. [CE, p. 168]). Let G and G0 be groups. A function φ: G0 G → is a crossed homomorphism if there is some homomorphism σ : G0 Aut G,such that the function → σ φ σ φ: G0 Aut G n G: g g ,g × → 7→ is a homomorphism. It is easy to check that σ φ is a homomorphism iff × φ φ hσ φ (gh) =(g ) h , g,h G0. ∀ ∈ In particular, every homomorphism is a crossed homomorphism, with σ being the trivial homomorphism (hσ = e). For a subgroup Γ0 of G0, we say that a crossed homomorphism φ is Γ0-equivariant if (gγ)φ = gφγφ, for every g G0 and γ Γ0. In other words, the associated homomorphism σ : G0 ∈ ∈ σ → Aut G is trivial on Γ0,thatis,(Γ0) =e. In particular, every homomorphism is Γ0-equivariant, for every subgroup Γ0. We now generalize the definition of an affine map, by allowing crossed homo- morphisms, instead of only allowing homomorphisms. The catch is that, to ensure n that a map from G0/Γ0 to R /Γ is well defined, we assume Γ0-equivariance. n Definition 1.4 (cf. Definition 1.3). For any choice of a basepoint x0 R ,the n n ∈ action of G on R provides a continuous map G R : g x0g. Then, if φ: G0 G is any crossed homomorphism, we have a composite→ map7→ → φ g x0g n φ0 : G0 G 7→ R , −−−−→ −−−−→ φ which is a crossed affine map. If, in addition, (Γ0) Γ, and φ is Γ0-equivariant, ⊂n then φ0 induces a well-defined map φ00 : G0/Γ0 R /Γ, and φ00 is also called a crossed affine map. → Main Theorem 7.20. Let Γ be a properly discontinuous subgroup of a connected, n n transitive, solvable Lie group of diffeomorphisms of R .IfR/Γis compact, then there is some discrete subgroup Γ0 of some simply connected, solvable Lie group G0, n and a crossed affine map φ: G0/Γ0 R /Γ, such that φ is a covering map, with finite fibers.
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