Theorem 9.1. For each large t, the horosphere quotient Ht/Γ admits an N -structure that has an orbit Ot such that the normal bundle to Ot is a flat Euclidean vector bundle with total space diffeomorphic to Ht/Γ.
248 i. belegradek and v. kapovitch
Arguing by contradiction, it suffices to prove the theorem for any sequence ti!
∞ such that Hti/Γ converges in pointed Gromov–Hausdorff topology. We fix such a sequence and assume for the rest of the proof that t belongs to the sequence.
We denote by gt the C1-Riemannian metric on the horosphere quotient Ht/Γ in-duced by the ambient metric (M, g) . Fix a small positive ε>0 to be determined later;
this constant will only depend on (M, g) . By Theorem 7.1, [14] and [28], for all large t there exists an N-structure on Ht/Γ with an orbit Ot such that the normal bundle to Ot is diffeomorphic to Ht/Γ . Also all orbits of the N-structure have diameter <ε with respect to an invariant metric ht that is ε-close to gt in uniform C1-topology, i.e.
|gt−ht|<ε and|∇gt−∇ht|<ε. It remains to show that for all larget, the normal bundle to Ot in Ht/Γ is flat Euclidean. The proof breaks into two independent parts.
In§9.1 we find a stratum Ft of the N-structure on Ht/Γ such that Ft is an ht -totally-geodesic closed submanifold that contains Ot with flat normal bundle. This uses only general properties of N-structures.
In§9.2 we show that the restriction to Ot of the normal bundle of Ft in Ht/Γ is flat, for all large t. This uses the flat connection of§4 and the fact that Ot!Ht/Γ is a homotopy equivalence.
9.1. The normal bundle in a stratum is flat
Throughout§9.1 we suppress the index t, writing O in place of Ot, etc. Let V be the tubular neighborhood ofO that is sufficiently small so that all orbits inV have dimension
>dim(O) . Let Oe and Ve be their universal covers. According to [14, pp. 364–365], the group Iso(Ve) contains a connected (but not necessarily simply-connected) nilpotent subgroup N that stabilizes Oe, acts transitively on Oe, and also Λ=N∩π1(V) is a finite index subgroup of π1(V) and is a lattice in N. The following lemma is implicit in [14].
Lemma 9.2. The subgroup H of Iso(Ve) generated by N and π1(V) is closed, N is the identity component in H, and the index of N in H is finite.
Proof. Λ is a cocompact discrete subgroup ofN and also of its closure Nin Iso(Ve) . Since dim(N) and dim(N) are both equal to the cohomological dimension of Λ , we get N=N. Now let Λ0 be a maximal finite indexnormalsubgroup ofπ1(Ve) that is contained in Λ . Ifγ∈π1(V) , thenN∩γN γ−1 contains Λ0 as a cocompact discrete subgroup, thus, as before, dim(N) and dim(N∩γN γ−1) are both equal to the cohomological dimension of Λ0, so N=N∩γN γ−1, and N is normalized by π1(V) . Thus N is normal in H, and Λ=Λ0. Since N is connected, it remains to show that |H:N| is finite. Since N and π1(V) generate H, the finite subgroup π1(V)/Λ of H/N generates H/N, hence π1(V)/Λ=H/N.
Let I be the intersection of the isotropy subgroups of H of the points of Oe. Since Oe is H-invariant, I is normal in H. The fixed point set of I is a totally geodesic submanifold Fe of Ve. The H-action on Fe descends to an H/I-action on Fe. Since π1(V) is torsion free and discrete, π1(V)∩I is trivial, and we identify π1(V) with its image in H/I. Denote the projection of Fe into V by F.
Lemma 9.3. The normal bundle to O in F is flat.
Proof. The group N acts transitively on Oe, so all isotropy subgroups for the N -action on Oe are conjugate. Since they are also compact, they lie in the center of N [20];
in particular, all the isotropy subgroups are equal, and hence each of them is equal to I∩N. In particular, N/(I∩N) acts freely and transitively on Oe. Since Oe is simply-connected, so isN/(I∩N) . Thus, I∩N is the maximal compact subgroup of N; hence, by Lemma A.3, I∩N is a torus, which we denote by T. The torus is the identity component of the compact group I, because |I:I∩N|6|H:N|<∞. Since N/T acts freely and transitively on Oe, we can choose a trivialization of ν, the normal bundle of F in Ve that is invariant under the left translations by N/T. Namely, let e∈Oe be the point corresponding to 1∈N/T under the diffeomorphism N/T∼=Oe. Fix an isomorphism φ: νe!{e}×Rk, and then extend it to the N/T-left-invariant isomorphism ν∼=O×Re k. Now, take γ∈π1(V) and x∈Oe. Using the above trivialization we define the rotational part of γ as the automorphism of {e}×Rk given by
φdLγ(x)−1dγdLxφ−1,
wheredLxis the differential of the left translation byx∈N/T. SinceO is an infranilman-ifold, π1(V) acts on Oe by affine transformations, that is if y∈Oe, then γ(y)=nγ·Aγ(y) , where nγ∈N/T and Aγ is a Lie group automorphism of N/T. Hence, for z∈Oe, we get
(Lγ(x)−1γLx)(z) =γ(x)−1·γ(xz) =Aγ(x)−1·n−1γ ·nγ·Aγ(xz) =Aγ(z) =Ln−1 γ γ(z), where the third equality holds as A is an automorphism and N/T=Oe. This establishes the above equality only on Oe, not on Fe, but both sides of the equality make sense as elements of H/I, and since Fe is the fixed point set of I, any two elements of H/I that coincide on Oe must coincide on Fe. Now the right-hand side is independent of x, which implies that the rotational part of γ is independent of x. This means that the bundle (Oe×Rk)/π1(V) is a flat O(k) -bundle, and hence so is the normal bundle ofF in V.
9.2. The normal bundle to a stratum is flat Let Ft be the stratum from§9.1.
250 i. belegradek and v. kapovitch
Lemma 9.4. For all large t, the restriction to Ot of the normal bundle to Ft in Ht/Γ is flat.
Proof. Let νt be the restriction to Ot of the normal bundle to Ft in Ht/Γ . By an obvious contradiction argument, it suffices to show that any subsequence of {ti}i has a subsequence for which the νt’s are flat; thus passing to subsequences during the proof causes no loss of generality. Since kt=dim(νt) can take only finitely many values, we pass to a subsequence for which kt is constant; we then denote kt by k. We look at the Grassmanian Gk(T M) of k-planes in T M with the metric induced by g. Of course, for k-planes tangent to Ht/Γ the metric is also induced by gt. We fix a point ot∈Ot and denote by Gkt the fiber of Gk(T M) over ot. Let lt be the fiber of νt over ot.
The fibers of Gk(T M) are (noncanonically) pairwise isometric via the Levi–Civita parallel transport of (M, g) . Let %diam(Gkt) be such that the center of mass of [22]
is defined in any 4%-ball of Gkt. For large enough t, the ball B(lt,4%)⊂Gkt contains no k-plane tangent to Ft because %diam(Gkt) , whereas lt and T Ft are ht-orthogonal, and ht and gt are ε-close, so that the distance between lt and any k-plane in T Ft is within ε of diam(Gkt) .
Denote by ∇∞ the connection from §4 associated with the parallel transport P∞. Also denote by ∇t the Levi–Civita connection of ht, and let Pt be its parallel transport.
The second fundamental form of Ht is uniformly bounded, hence P∞ and Pt are close over any short loop in Ht/Γ .
Since ∇∞ is flat and compatible with the metric g, it defines the holonomy homo-morphism φt: Γ!Iso(TotM) . Let Rt be the closure of φt(Γ) in Iso(TotM) . Since Rt is compact, there exists afinite subset St⊂Γ so that for each rt∈Rt there exists s∈St such that rt(lt) and φt(s)(lt) are %-close in Gkt. The direction orthogonal to Ht/Γ is
∇∞-parallel, so the Rt-action preserves the subspace T Ht/Γ⊂T M. Also P∞ defines an isometry between the Gkt’s that is φt-equivariant and Rt-equivariant. Thus St can be chosen independently of t, and we denote St by S.
If t is sufficiently large, then, for every s∈S, the k-planes φt(s)(lt) and lt are
%-close in Gkt. Indeed, by the exponential convergence of geodesics, if t is large, then s can be represented by a short loop based at ot. Since the inclusion Ot,!Ht/Γ induces a homotopy equivalence, the loop can be assumed to lie on Ot⊂Ft. Since Ft is ht-totally geodesic and lt is orthogonal toFt and has complementary dimension,lt is fixed by Pt along any loop in Ft based at ot. Hence the same is almost true for P∞, provided the loop is short enough, which proves the claim since S is finite.
Thus, by the triangle inequality in Gkt, the k-planes rt(lt) and lt are 2%-close for all rt∈Rt. Now let ¯lt be the center of mass of all rt(lt) ’s with rt∈Rt. Clearly, ¯lt is 2%-close to lt, hence ¯lk is transverse to T Ft. Since P∞ preserves T Ht/Γ , each rt(lt) ,
and hence ¯lt, is tangent to Ht/Γ . Now we translate ¯lt around Ot using P∞ along paths of length 6diam(Ot)<ε. Since ¯lt is Rt-invariant and ∇∞ is flat, this gives a well-definedk-dimensional flat C0 subbundle ¯νt of the restriction ofT Ht/Γ toOt⊂Ft. Note that Pt takes any k-plane tangent to Ft to a k-plane tangent to Ft, since Ft is totally geodesic. On the other hand, ¯ltis a uniform distance away from any k-plane in T Ft, so its Pt-image remains far away from any k-plane in T Ft. Since on short paths Pt is close to P∞, we conclude that ¯νt is transverse to T Ft for large t. Thus ¯νt is C0-isomorphic to νt, so that νt carries a C0-flat connection.
In fact, νt also carries a smooth flat connection. Indeed, in the universal cover Oet, the C0-flat connection defines a Γ -equivariant C0-isomorphism of the pullback of νt to OetontoOet×Rk, where Γ acts as the covering group on the Oet-factor and via a holonomy homomorphism on the Rk-factor. This is a smooth action, so the quotient (Oet×Rk)/Γ is a smooth flat vector bundle that is C0-isomorphic to νt. But any C0-isomorphic bundles are smoothly isomorphic because the continuous homotopy of classifying maps can be approximated by a smooth homotopy.