aa r X i v : . [ m a t h . G T ] F e b SYSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS JOE SCULL
Abstract.
We show that the length R of a systole of a closed hyperbolic n -manifold ( n ≥
3) admitting a triangulation by t n -simplices can be boundedbelow by a function of n and t , namely R ≥ ( nt ) O ( n t ) . We do this by finding a relation between the number of n -simplices and the di-ameter of the manifold and by giving explicit bounds for a well known relationbetween the length of the core curve of a Margulis tube and its radius.We prove the same result for finite volume manifolds, with a similar butslightly more involved proof. Introduction
A major tool used to understand Riemannian manifolds is examining how dif-ferent geometric properties of such manifolds relate to one another or converselyidentifying when there is no such relation. Gromov [Gro78] proves a fundamentalrelation between volume and diameter for Riemannian manifolds M of negativecurvature and dimension n ≥
8, proving that if sectional curvature of M lies in theinterval ( − , vol ( M ) ≥ C n (1 + diam ( M ))where C n is some constant dependent on n . A similar but slightly weaker result isalso proven for n ≥
4. Conversely, such a result cannot hold in dimension 3 as thereexist sequences of hyperbolic 3-manifolds with volume uniformly bounded whosediameters tend to infinity, a fact which follows from Thurston’s hyperbolic Dehnsurgery theorem [Thu78]. Influential to Gromov’s paper was one almost a decadeearlier by Cheeger [Che70] in which he finds a lower bound on injectivity radiusas a function of volume, diameter and a lower bound on sectional curvature. Hegoes on to use this relationship to prove the famous Cheeger finiteness theorem, amodern statement of which can be found in [Cha95].For hyperbolic manifolds of dimension ≥
3, Margulis’ Lemma and the resultingthick-thin decomposition tell us that having a sufficiently short systole is indicativeof the existence of Margulis tubes, whose diameters grow as the length of their corecurves decrease. In this paper we give an explicit lower bound on how the diameterof these tubes relate to the length of the core curve, and further show how thisdepends on n . Mathematics Subject Classification.
Key words and phrases.
Injectivity Radius, Systole, Hyperbolic Manifold, Triangulation.This work was supported by The Engineering and Physical Sciences Research Council (EPSRC)under grant EP/R513295/1 studentship 2100094.
Theorem 4.8.
Suppose that M is a finite volume hyperbolic n -manifold ( n ≥ with systole(s) of length R ≤ ǫ n , where ǫ n is the Margulis constant in dimension n . Then the distance from a systole to the boundary of the Margulis tube containingit is bounded below by n log (cid:18) R (cid:19) + log ( ǫ n ) − log (4) . A less explicit version of this theorem is surely known to experts. In dimensions ≥ n ≥ Theorem 1.1.
Given a closed (or finite volume) hyperbolic n -manifold ( n ≥ triangulated by t n -simplices (possibly semi-ideal), the length R of a systole of M is bounded below by a function of t and n , in particular R ≥ ( nt ) O ( n t ) Remark.
Note that for a non-orientable hyperbolic n -manifold, we can apply The-orem 1.1 to its orientable double cover and use that to bound the length of systolesin the original manifold. In doing so we double the number of tetrahedra and halvethe bound on systole length. This difference disappears when we simplify big Onotation however.Because of this, we shall from this point on assume all our manifolds to beorientable, and thus all our lattices shall lie in Isom + ( H n ).It’s worth noting that at first, one might think this theorem for n ≥ n inan unknown way.In dimension 3 Theorem 1.1 forms a crucial part of the argument in an upcomingpaper of the author [Scu21] on a bounded runtime algorithm for the homeomor-phism problem for hyperbolic 3-manifolds. That paper uses this same idea of findinga link between the combinatorial information of a triangulation and geometric in-formation about the manifold and both papers are inspired by Kuperberg’s work[Kup19] and his use of the results of Grigoriev and Vorobjov.In fact in dimension 3 a lot more is known. Futer, Purcell and Schleimer [FPS19]prove a 3-dimensional version of Theorem 4.8 which they prove to be sharp up tothe additive constants and which is still linear in log(1 /R ). Kalelkar and Raghunath[KR20] also provide a result linking the number of tetrahedra in a triangulation ofa cusped hyperbolic 3-manifold M to systole length. Their result gives much betterbounds in terms of t but requires that the original triangulation be geometric andalso depends on the geometry of the triangulation, which is quite a strong restric-tion. A closed 3-dimensional version of Theorem 1.1 follows from a combination YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 3 of Theorem 4.8 (or indeed [FPS19]) and another result relating combinatorial andgeometric properties of hyperbolic 3-manifolds, due to Matthew White [Whi00a](see also [Whi00b] [Whi02]). Theorem (Theorem 5.9, [Whi00b]) . There is an explicit constant
K > such thatif M is a closed, connected, hyperbolic 3-manifold, and P = h x , . . . , x n | r , . . . , r m i is a presentation of its fundamental group, then diam ( M ) < K ( l ( P )) , where l ( P ) = m X i =1 l ( r i ) and l ( r i ) is the word length of a given relator. The 3-dimensional version of Theorem 1.1 that follows from this combinationhas a tighter bound on systole length of the form R ≥ O ( t ) . The proof follows from considering a natural presentation of the fundamental groupinduced by the given triangulation. For further examples of low-dimensional resultsin this area, see Adams and Reid [AR00], Kalelkar and Phanse [KP19] or theaforementioned work of Agol [Ago06].It’s also worth noting that in dimension two such a result does not hold, in facta given surface of negative euler characteristic admits hyperbolic structures witharbitrarily small curves. This is shown in any introductory text on Teichm¨ullerspace, see for example [FM11] [Mar16].1.1.
Sketch of the Paper.
In Section 2 we recount the work of Grigoriev andVorobjov which bounds the size of a solution to a system of polynomials as afunction of the size of the system. In Section 3 we use the fact that cocycles into
Isom + ( H n ) are solutions to some natural system of polynomials and the results ofSection 2 to find a cocycle which is “bounded”. This cocycle corresponds to somefaithful lattice representation of the fundamental group of a given n -manifold whichadmits a hyperbolic structure. We can then use this cocycle to define a homotopyequivalence from a triangulation of this manifold to some specific hyperbolic n -manifold X . This homotopy equivalence has as its image a union of geometricsimplices, and the bounds on the cocycle control the geometry of these simplices,in particular their diameters are bounded.In Section 4 we use Margulis’ Lemma to calculate a relationship between asystole’s length and the diameter of the Margulis tube containing the systole. Wethen use the bounds on the geometry of the simplices to bound the diameter of aclosed hyperbolic manifold X and hence to bound the length of its systoles.In Section 5 we adapt some of the results used above to the finite volume caseand apply the same result on Margulis tubes. To do this we find a bound on thediameter of a subset of the manifold which contains all the Margulis tubes.2. Systems of Polynomial Inequalities
What we present here is adapted from [Gri86] (which is a survey of the resultsin [GV88]) with a few corollaries which show how to adapt the theorem to our
JOE SCULL situation. This section shows how given a system of polynomial inequalities, onecan find solutions which are bounded in terms of the ‘size’ of the system.Let a system of polynomial inequalities f > , . . . f m > , f m +1 ≥ , . . . , f κ ≥ f i ∈ Q [ X , . . . , X N ] satisfy the bounds deg X ,...,X N ( f i ) < d, l ( f i ) < M, ≤ i ≤ κ. Here deg X ,...,X N ( f i ) is the maximum degree of a monomial in f i , where the degreeof a monomial is the sum of the exponents of the monomial. The length, or com-plexity l is defined on rational numbers by l ( pq ) = log ( | pq | + 2) and on polynomialsit is defined as the maximum complexity among its coefficients.Let α = ( α , . . . , α N ) be a solution to the system of inequalities where each α i isan algebraic number. Then then we can represent each α i in terms of a primitiveelement, θ , of the field Q ( α , . . . , α N ) = Q ( θ ). We represent θ by providing anirreducible polynomial Φ( X ) ∈ Q ( X ) of which θ is a root and an interval ( β , β ) ⊆ Q with endpoints in Q which determines θ among the roots of Φ. With θ defined,one has α i = P j α ( j ) i θ j for α ( j ) i ∈ Q . Using this notation we can now state thetheorem of Grigoriev. Theorem 2.1 (Grigoriev [Gri86]) . For a given system of inequalities such as thoseabove, for each connected component of the solution set, there exists a solution α = ( α , . . . , α N ) represented as above for which the following are true: deg (Φ) ≤ ( κd ) O ( N ) ; l (Φ) , l ( α ( i ) j ) , l ( β ) , l ( β ) ≤ M ( κd ) O ( N ) The following corollary follows the same method of proof as that of Lemma 8.9in [Kup19], but for self-containedness, we provide it here.
Corollary 2.2. M ( κd ) O ( N ) ≤ | θ | ≤ M ( κd ) O ( N ) Proof.
Note that we can scale Φ so that it is an integer polynomial without negatingthe statement that l (Φ) ≤ M ( κd ) O ( N ) . So we may assume that the coefficients, γ i of Φ are integers, and hence the γ i have size bounded by 2 l (Φ) − θ ) = 0 we get that − γ deg (Φ) θ deg (Φ) = γ + . . . + γ deg (Φ) − θ n − and so | θ | = (cid:12)(cid:12)(cid:12)(cid:12) γ + . . . + γ deg (Φ) − θ n − θ n − γ deg (Φ) (cid:12)(cid:12)(cid:12)(cid:12) ≤ deg (Φ) − X i =0 | γ i | Note that θ − is a root ofΨ( x ) = x n γ + . . . + γ deg (Φ) = x n Φ( x − )and so 1 | θ | = | θ − | ≤ deg (Φ) X i =0 | γ i | . Applying the bounds on the γ i gives | θ | ≤ deg (Φ) − X i =0 γ i ≤ deg (Φ) − X i =0 l (Φ) ≤ deg (Φ)2 l (Φ)YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 5 and similarly, | θ | ≥ deg (Φ)2 l (Φ) Finally, Theorem 2.1 tells us that deg (Φ)2 l (Φ) ≤ ( κd ) O ( N ) M ( κd ) O ( N ) ≤ M ( κd ) O ( N ) and the statement follows. (cid:3) From this, we can derive a statement about the size of the solutions themselves.
Corollary 2.3.
Let α be as in Theorem 2.1, then for each i , if α i = 012 M (( κ +2 N ) d ) O ( N ) ≤ | α i | ≤ M ( κd ) O ( N ) . Proof.
First see that we can achieve an upper bound on | α i | by applying the triangleinequality in the following manner: | α i | ≤ | deg (Φ) X j =0 α ( j ) i θ j | ≤ deg (Φ) X j =0 | α ( j ) i || θ j | ≤ deg (Φ) X j =0 M ( κd ) O ( N ) M ( κd ) O ( N ) ≤ (( κd ) O ( N ) + 1)2 M ( κd ) O ( N ) M ( κd ) O ( N ) ≤ M ( κd ) O ( N ) . Note that a lower bound on | α i | is equivalent to an upper bound on | α − i | , so we canmodify our system of polynomial inequalities by adding some β i such that α i β i = 1for all i . This is represented by two inequalities and one new variable for each ofthe original variables. Now applying the theorem to this new system of inequalitiesgives an upper bound for | β i | as above except κ is replaced by κ + 2 N , as O (2 N )and O ( N ) are the same. Taking reciprocals of both sides gives us the lower boundin the statement. (cid:3) Before we move on we should note that the theorem requires that the systembe given as polynomial inequalities. When we create our system, we shall allow forequalities as well, which can be replaced by a pair of inequalities. Because of big Onotation, doubling the number of polynomials will not affect the final bound andso we shall ignore this distinction from here on.3.
Cocycles and Homotopy Equivalences
Definition 3.1 (Triangulation) . A triangulation of an n -manifold M is a simplicialcomplex equipped with a homeomorphism to M . Remark.
The results of this paper also hold for a more general definition of tri-angulation often used by low dimensional topologists, at least in dimension 3. Inhigher dimensions, one tends to need to introduce further requirements to avoidpathologies, and so we shall stick to the simplicial setting. For a more detaileddiscussion see Thurston’s definition of a rectilinear gluing and further comments in[Thu97]. In support of this remark efforts have been made throughout the paperto avoid the use of certain properties of simplicial complexes that aren’t shared bymore general triangulations, for example that two n -simplices intersect in at mostone face. JOE SCULL
Definition 3.2 (Star and Link) . The star st(∆) of a simplex ∆ in a simplicialcomplex K is the smallest subcomplex of K containing all simplices which have ∆as a face. The link lk(∆) of a simplex ∆ in a simplicial complex K consists of allsimplices in st(∆) which do not contain ∆. Definition 3.3 (Simplicial Path) . A sequence of oriented 1-simplices in a trian-gulation such that the end vertex of each simplex is the start vertex of the next iscalled a simplicial path.
Definition 3.4.
Let M be a triangulated n -manifold and Λ a group. A cocycle α ∈ C ( M, Λ) is a map from the oriented 1-simplices of M to Λ which satisfies thefollowing conditions • For each 2-simplex of M , let a, b, c be the three edges of the simplex orientedsuch that a and c start at the same vertex, and b and c end at the samevertex, then α ( a ) α ( b ) = α ( c ) • if ¯ a is the same edge as a with the opposite orientation then α ( a ) − = α (¯ a ) Remark.
We can extend α to a map from all simplicial paths γ = e . . . e n bydefining α ( γ ) = α ( e ) . . . α ( e n )and then because every homotopy of simplicial paths can be realised as a sequenceof homotopies across 2-simplices, α is invariant under homotopy fixing end points.This means that α induces a homomorphism α ∗ : π ( M ) → Λ. Definition 3.5 (Semi-ideal n -simplex) . A semi-ideal n -simplex is one which hashad one of its vertices removed. We call this vertex ideal. Definition 3.6 (Semi-Ideal Triangulation) . Let M be a finite volume hyperbolic n -manifold, then a semi-ideal triangulation of M is a simplicial complex where weremove some of the vertices before defining a homeomorphism to M . Convention.
In the case of a semi-ideal triangulation of a manifold M , a cocycle α ∈ C ( M, Λ) shall take as inputs only the non-ideal edges of the triangulation.This is justified as the semi-ideal triangulation deformation retracts to its non-idealsimplices.
Definition 3.7 (G-equivariant) . Let
X, Y be G -sets for G a group. Let F : X → Y be a map such that ∀ g ∈ G, ∀ x ∈ X, F ( g · x ) = g · F ( x )Then we say F is G -equivariant and F induces a map ˆ F : G \ X → G \ Y . Definition 3.8.
Let ˜ M be the universal cover of M and let φ be the canonicalisomorphism π ( M, x ) → Deck ( ˜
M , ˜ x ) defined by picking as a base point the lift˜ x of x , then for each [ γ ] ∈ π ( M, x ) we define g γ := φ ([ γ ]) .Let M be a n -manifold admitting a finite volume hyperbolic structure equippedwith a triangulation (semi-ideal if the manifold is not closed) and let X = G \ H n be a hyperbolic n -manifold homeomorphic to M , so G ≤ Isom + ( H n ). Let f bea homeomorphism f : M → X , then f induces a G -equivariant homeomorphism˜ f : ˜ M → H n because the action of G on ˜ M is defined by following the chain ofisomorphisms G ∼ = π ( X ) f ∗ ←− π ( M ) ∼ = Deck ( ˜ M ) . YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 7 Lemma 3.9.
Let
M, X, G be as in the preceeding paragraph. Let α be a cocycle suchthat α ∗ : π ( M, x ) → Isom + ( H n ) is a representation of π ( M ) with Im ( α ∗ ) = G .Then we can define a map ˜ F : ˜ M → H n such that for each (possibly semi-ideal) n -simplex ∆ in ˜ M , ˜ F (∆) is a geodesic n -simplex (it is the convex hull of its vertices)and ˜ F induces a homotopy equivalence F : M → X . Furthermore, the map which F induces on fundamental groups is exactly α ∗ .Proof. First suppose that M is closed. Let ˜ x ∈ ˜ M be some lift of x and take˜ F ( ˜ x ) = ˜ y some arbitrary point in H n . Now let ˜ x be any other vertex in the0-skeleton of ˜ M and ˜ γ a simplicial path from ˜ x to ˜ x . Let γ be the projection of˜ γ to M , then define ˜ F (˜ x ) = α ( γ ) · ˜ y . Assuming that ˜ F is G -equivariant on thevertices, we define ˜ F dimension by dimension, first we map the 1-simplices param-eterised by arc-length onto the geodesic between the image of their vertices. Forhigher dimensional simplices, pick one representative of each G -orbit, and defineinductively on dimension by picking for each k -simplex representative some home-omorphism onto the convex hull of the image of its vertices which is an extensionof the homeomorphism defined inductively on its boundary. We then define ˜ F onall the translates of our simplex by ˜ F ( g ( x )) = α ∗ ( g ) ˜ F ( x ) for all x in our chosenrepresentative.By construction, ˜ F is G -equivariant iff the map on the 0-skeleton is. Let ˜ x beas above, then ˜ x = g γ ( ˜ x ). Take g γ ′ some other element of Deck( ˜ M ), then˜ F ( g γ ′ (˜ x )) = ˜ F ( g γ ′ g γ ( ˜ x )) = α ( γ ′ ) α ( γ ) ˜ F ( x ) = α ( γ ′ ) ˜ F ( g γ ( ˜ x )) = α ( γ ′ ) ˜ F (˜ x ) . Hence, ˜ F is G -equivariant.Note here that if we take some homeomorphism from M to X , then this lifts toa map ˜ F ′ from ˜ M to ˜ X and by point-pushing G -equivariantly, we can assume thatthis map agrees with ˜ F on the vertices. Because of this we can define a straightline homotopy on each simplex of ˜ M which commutes with the G -action and takes˜ F to ˜ F ′ . Thus F and F ′ are homotopic, and as F ′ is a homeomorphism, F mustbe a homotopy equivalence.In the cusped case, we can use the same argument if we can define ˜ F on theideal vertices in ˜ M and show it to be G -equivariant on them too. Note here thatwe are in fact defining ˜ F by defining a map from ˜ M ∪ { ideal vertices } to H n ∪ ∂ H n and then taking its restriction to ˜ M . However, we will abuse notation to refer toboth maps as ˜ F throughout. We number the cusps and take a spanning tree of the1-skeleton of the link of the ideal vertex v i corresponding to the i th cusp and addsome edge which ends at v i . We also choose a path δ in the 1-skeleton of M fromour basepoint to this tree such that the union is still a tree, call this union Γ i . Anyedge in the link of the ideal vertex which is outside of the spanning tree gives aloop σ by adjoining the unique geodesic in Γ i linking its endpoints. We can make σ be based at x by adjoining our path δ at both ends. As σ is homotopic into theideal vertex, α ( σ ) is thus a parabolic element of G , fixing some point p in ∂ H n .If ˜ v i is the lift of v i which lies in the same lift of Γ i as ˜ x , then we define˜ F ( ˜ v i ) = p . Each lift ˜ v i of v i lies in a lift of Γ i so if we take some path γ from ˜ x to this new lift of Γ i , then we define ˜ F ( ˜ v i ) = α ( γ ) p . From here, we only need tocheck two things, first that ˜ F is well defined, and second that it is G -equivariant.For the well definedness, note that for a lift ˜ v i , there are many lifts of Γ i to whichit might belong and thus distinct choices, γ, γ ′ for the path defined above. However JOE SCULL all such pairs of lifts of Γ i can be connected by a path which lies entirely in the linkof the lift and maps down to some loop δ which is homotopic into v i . Thus α ( δ )corresponds to a parabolic element fixing p and so as γ ′ = γδ , we get that α ( γ ′ ) p = α ( γ ) α ( δ ) p = α ( γ ) p so ˜ F is well defined. For the G -equivariance, note that for g some deck transforma-tion ˜ x and g ( ˜ x ) are linked by a path γ with α ( γ ) = g so as g ˜ v i lies in the samelift of Γ as g ˜ x we have that˜ F ( g ˜ v i ) = α ( γ ) p = α ( γ ) ˜ F ( ˜ v i ) = g ˜ F ( ˜ v i )and hence for any h ∈ G and any ideal vertex ˜ v = g ˜ v i ∈ ˜ M , the following holds˜ F ( h ˜ v ) = ˜ F ( hg ˜ v i ) = hg ˜ F ( ˜ v i ) = h ˜ F (˜ v ) (cid:3) Lemma 3.10.
A homotopy equivalence f : M → N between closed orientable n -manifolds is a surjection.Proof. Note that as f is a homotopy equivalence, it induces an isomorphism on n thhomology groups, sending a fundamental class [ M ] to a fundamental class [ N ].Now suppose f is not surjective, and let y ∈ N be outside the image of f . Thenconsider the natural isomorphism H n ( N ) ∼ = H n ( N, N − y ) then for any representa-tive of [ M ], f [ M ] lies entirely in N − y and so f [ M ] is trivial in H n ( N, N − y ) andthus trivial in H n ( N ) contradicting that f is a homotopy equivalence. (cid:3) Remark.
It is also possible (see Lemma 5.1) to prove that the homotopy equivalencedefined in Lemma 3.9 is surjective in the non-compact case, but that requires a morenuanced argument as homotopy equivalences between non-compact manifolds caneasily not be surjections, for example consider a standard map from R n to itselfwhich maps everything to the point.We will soon see how to find cocycles as the solutions of a system of polynomials,but these cocycles could be trivial or induce non faithful representations, so to usethe results of Section 2 we need some connected component of the solution setto consist of cocycles which all induce faithful lattice representations (see belowdefinition) which define homotopy equivalences by the method of Lemma 3.9. Weshall prove the existence of this connected component using Calabi-Weil Rigidity. Definition 3.11.
Let G be a group, Λ = Isom ( H n ) and let H := { H , . . . H q } acollection of subgroups of G . Define Hom par ( G, H ; Λ) := { ρ : G → Λ | ρ ( γ ) is parabolic for all γ ∈ H j , j = 1 , . . . q } If G is a subgroup of Λ, then we define Hom par ( G ; Λ) := Hom par ( G, H max ; Λ)where H max is a collection of representatives of the conjugacy classes of maximalparabolic subgroups of G . Note that if G is a uniform lattice, then Hom par ( G ; Λ) = Hom ( G ; Λ). Theorem 3.12 (Calabi-Weil Rigidity, Thms. 4.19 and 8.55 in [Kap00] ) . Let G be a lattice in Isom ( H n ) where n ≥ . The identity representation ρ : G → Isom ( H n ) := Λ lies in a connected component of Hom par ( G ; Λ) made up entirelyof conjugates of ρ . YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 9 For the sake of comparison we place here the similar result of [GR70] which weshall use for the cusped case in section 5. It crucially doesn’t cover 3-dimensionalmanifolds so we can’t use it in the closed case, but it avoids the requirement onparabolics in the cusped case.
Theorem 5.2 (Theorem 7.2 in [GR70]) . Let G be a lattice in Isom ( H n ) for n ≥ .The identity representation ρ : G → Isom ( H n ) := Λ lies in a connected componentof Hom ( G ; Λ) made up entirely of conjugates of ρ . Definition 3.13 (Lattice Representation) . We call a representation G → Isom ( H n )a lattice representation if its image is a lattice (a discrete co-finite volume subgroup).To round out our rigidity results, we recall Mostow-Prasad rigidity, which tellsus (among other things) that all faithful lattice representations of a group induceisometric hyperbolic structures, a fact that we shall be implicitly using throughoutthe rest of the paper. Theorem 3.14 (Mostow-Prasad Rigidity [Pra73]) . Let Γ , Γ ′ be two lattices in Isom ( H n ) and φ : Γ → Γ ′ an isomorphism . Then there exists an isometry g in Isom ( H n ) such that gγg − = φ ( γ ) ∀ γ ∈ ΓIn the following lemma we use the isomorphism SO ( n, ∼ = Isom + ( H n ) comingfrom the hyperboloid model of hyperbolic n -space. Lemma 3.15.
Let M be a closed hyperbolic n -manifold triangulated by t n -simplices.There is a system of polynomial inequalities as in Theorem 2.1 such that the fol-lowing all hold: • The solution set consists of cocycles in C ( M, SO ( n, . • The solution set has a component consisting entirely of cocycles each ofwhich induces a faithful lattice representation. • The system consists of ≤ ( n + 1) t polynomials, in ≤ ( n + 1) t variableswith degree ≤ and coefficients all ± .Proof. For each edge e of the triangulation of M (of which there are less than (cid:0) n +12 (cid:1) t ), define two sets of ( n + 1) variables corresponding to the entries of amatrix in SO ( n, e under some cocycle α . Then, for each dimension 2 face of the triangulation(of which there are less than (cid:0) n +13 (cid:1) t ) we get a relation which can be expressed as( n + 1) polynomials (for the ( n + 1) variables defining the matrix) which aresatisfied iff the matrices corresponding to the first two edges of the face multiply togive the matrix corresponding to the third edge, these being the face relations ofthe cocycle. We also include polynomials which ensure that opposite orientationsof an edge are sent to inverse elements, this consists of ( n + 1) polynomials foreach edge, and these polynomials have coefficients ± ≤ ( n + 1) t polynomials in ≤ ( n + 1) t variableswith coefficients always 1 or -1 and which are at most quadratic. Furthermore,solutions to this system correspond to maps from the oriented 1-simplices of M to SO ( n,
1) which satisfy the face relations and orientation reversing relations, thusby definition solutions correspond to 1-cocycles.As our manifold M admits a hyperbolic structure, we know that it admits afaithful lattice representation and thus there exists a cocycle which induces such a representation. Now any continuous deformation of this cocycle induces a continu-ous deformation of the induced representation. Thus Calabi-Weil Rigidity impliesthat any component containing a cocycle which induces a faithful lattice represen-tation consists entirely of such cocycles. (cid:3) At this point we could apply the results of Section 2 and get bounds on the sizeof the matrices in our cocycle and use that to find bounds on the geometry of M .Instead, we’re going to expand the system of polynomials so that the geometricquantities we want to bound appear among the variables and thus the results ofSection 2 immediately give us the desired bounds.For the following Lemma it is useful to note that by finiteness of the triangulation,any cocycle can be altered (by multiplying wherever necessary the value on alledges round a vertex by a sufficiently small isometry) so as to induce the samerepresentation, while taking non-zero values on every edge. Thus we don’t lose anyrepresentations by making this requirement of our cocycles. Lemma 3.16.
We can expand the system of polynomials above to include variablescorresponding to the lengths of the edges in the image of the homotopy equivalence F defined in Lemma 3.9. We also add the requirement that each edge be non-zerolength.The solutions of this new system are in one to one correspondence with the subsetof the old solution set consisting of all cocycles which take non-zero values on alledges.Proof. In the proof of Lemma 3.9 we didn’t specify a model for the universal coverof either manifold, though both have universal cover H n . Here we shall take thehyperboloid model as our model and we shall take (0 , . . . , ,
1) as our basepoint.Now SO ( n,
1) acts on the hyperboloid by matrix multiplication. Thus the map onuniversal covers ˜ F : ( ˜ M , ˜ x ) → ( H n , (0 , . . . , , γ in ˜ M to the point A γ (0 , . . . , , T where α ( γ ) =: A γ ∈ SO ( n, A (0 , . . . , , T where A is A γ for some γ .As there are at most (cid:0) n +12 (cid:1) t edges in M we can find a lift ˜ e of each edge e of M such that both vertices x, y of ˜ e lie within (cid:0) n +12 (cid:1) t edges of the basepoint. Pickingpaths γ , γ of length ≤ (cid:0) n +12 (cid:1) t to each vertex of ˜ e , yields group elements A γ , A γ which can be expressed as ( n + 1) polynomials of degree at most (cid:0) n +12 (cid:1) t in theoriginal variables (each A γ is a product of A e i with e i edges of the triangulation).We build this up edge by edge, at each point we define at most one new vertex foreach new edge as a set of n + 1 new variables. In total we need ≤ ( n + 1) (cid:0) n +12 (cid:1) t newvariables and polynomials defining these vertices. The length of the edge betweentwo such vertices x and y where x = ( x , . . . , x n ) and y = ( y , . . . , y n )can then becalculated as l ( e ) = d ( x, y ) = arcosh ( x y + . . . + x n − y n − − x n y n ) . This is not a polynomial and so we instead define a variable for each edge as C = cosh ( l ( e )) − cosh ( d ( x, y )) − C is the number of edgeswhich is ≤ (cid:0) n +12 (cid:1) t . We then require that C >
0, and thus that l ( e ) > YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 11 In total we have added less than ( n + 2) (cid:0) n +12 (cid:1) t new variables and ( n + 2) (cid:0) n +12 (cid:1) t polynomials with degree bounded by (cid:0) n +12 (cid:1) t and coefficient complexity still at 1. (cid:3) Combining the above two proofs gives a polynomial system which has size de-scribed (in the language of Section 2) by • N ≤ ( n + 1) t + ( n + 2) (cid:0) n +12 (cid:1) t ≤ ( n + 2) t • κ ≤ ( n + 1) t + ( n + 2) (cid:0) n +12 (cid:1) t ≤ ( n + 2) t • d ≤ (cid:0) n +12 (cid:1) t ≤ ( n + 1) t • M ≤ Theorem 3.17.
Let M be a closed hyperbolic manifold, there exists a homotopyequivalence F as described in Lemma 3.9 such that the image of each edge has length l ( e ) bounded in the following way: cosh ( l ( e )) ≤ n +2) t ) O (( n +2)4 t ) This can be simplified, due to big O notation to the following bound: cosh ( l ( e )) ≤ ( nt ) O ( n t ) And hence, that < l ( e ) ≤ ( nt ) O ( n t ) Proof.
Lemmas 3.15 and 3.16 provide us with a system of polynomials, the solutionset of which has a connected component, all elements of which induce faithful latticerepresentations. Mostow Rigidity tells us each of these lattices defines a hyperbolicmanifold X as in the setup of Lemma 3.9. Thus taking a bounded solution givenby Theorem 2.1 means that all the non-zero variables of this solution are boundedas in the statement of the Theorem. In particular, as edge lengths are all non-zerothe result follows. (cid:3) Diameter of Margulis Tubes
In this section we shall describe some results of hyperbolic geometry which willallow us to bound the length of a systole from below as a function of the diameter ofthe Margulis tube containing it. This will be used in conjuction with our bound onedge lengths from the previous section to bound the systole length of a hyperbolicmanifold M from below as a function of the number of tetrahedra in any topologicaltriangulation of M .The tool we will use to do this is the thick-thin decomposition, which we shallbriefly recall here, following the books of Benedetti and Petronio and Jessica Purcell[BP92][Pur20]. Remark.
The model of hyperbolic space we’re using in this section is different fromin the previous section. For the purpose of understanding cocycles and defininga system of polynomials, we need to be working with a variety so we used thehyperboloid model because its isometry group is a variety. In this section, we willuse the Euclidean structure of horospheres and so we use the upper half spacemodel where any given horosphere can be taken to a horizontal Euclidean plane byan isometry of H n . Definition 4.1 (Injectivity Radius) . We define the injectivity radius at a point x ∈ M to be injrad( x ) = sup { r | B ( x, r ) is an embedded r-ball in M } . The injectivity radius of a manifold is then given byinjrad( M ) = inf { injrad( x ) | x ∈ M } . Note that in the case where M is closed, there is some x ∈ M such that injrad( x ) =injrad( M ). Definition 4.2.
A systole of a hyperbolic manifold M is a geodesic γ in M ofshortest length. Definition 4.3.
We define the ǫ -thin part M (0 ,ǫ ) of a hyperbolic manifold M to be M (0 ,ǫ ) = { x ∈ M | injrad( x ) < ǫ } . Similarly the ǫ -thick part is defined to be M [ ǫ, ∞ ) = { x ∈ M | injrad( x ) ≥ ǫ } Theorem 4.4 (Margulis’ Lemma, D3.3 in Benedetti and Petronio [BP92]) . Thereexists a universal constant ǫ n for each n ≥ such that the following holds.Let M be a complete oriented hyperbolic n -manifold (not necessarily compact orfinite volume). The thin part M (0 ,ǫ n ) is the union of pieces homeomorphic to oneof the following types: (1) ˚ D n − × S (2) V × (0 , ∞ ) where V is a differentiable oriented ( n − -manifold withoutboundary supporting a Euclidean structure.Furthermore, • The pieces are a positive distance from one another. • The second case occurs only when the manifold is non-compact. • When M is finite volume non-compact, the second case occurs and themanifolds V are closed. Definition 4.5.
We shall refer to the two types of component of the thin part astubes (also called Margulis tubes) and cusp neighbourhoods.Note that if a systole of a hyperbolic n -manifold lies in the thin part of M , itmust lie in one of these tubes. Otherwise it lies in a cusp neighbourhood, and couldbe homotoped further towards the cusp to reduce its length.We now consider what these Margulis tubes are like if a systole of the manifoldis very short. First a quick lemma about closest point projection to the verticalaxis in the upper half space model. Lemma 4.6.
The distance from a point x to the vertical axis through the origin inthe upper half space model is given by arcosh (cid:18) k x k x n (cid:19) where k x k is the standard Euclidean norm. YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 13 Proof.
Let x := ( x , . . . x n ) be a point in the upper half space model and note thatthe closest point projection of x to the vertical axis is y := (0 , . . . , , k x k ) as bothpoints lie on a half circle with centre on the boundary at infinity which intersectsthe vertical axis at a right angle.Now a formula for distance in the upper half space model is given by d ( x, y ) = arcosh (cid:18) P ni =1 ( x i − y i ) x n y n (cid:19) So substituting in our values gives us d ( x, y ) = arcosh (cid:16)P n − i =1 ( x i ) (cid:17) + ( x n − k x k ) x n k x k = arcosh (cid:18) k x k − x n k x k x n k x k (cid:19) = arcosh (cid:18) k x k x n (cid:19) (cid:3) Lemma 4.7.
Let φ be a loxodromic isometry in Isom ( H n ) acting on the upper halfspace model with fixed points at and ∞ . Suppose the translation length of φ alongits axis is R . If x is such that k x k x n ≤ e D , then for each > a > , we can find aninteger < k ≤ (cid:16) e D a (cid:17) n − s.t. d (cid:0) φ k ( x ) , e kR ( x ) (cid:1) < a .Proof. A loxodromic isometry φ of translation length R acting on the upper halfspace model of hyperbolic space fixing 0 and ∞ is a composition of a scaling x e R x and an orthogonal map A ∈ SO ( n −
1) acting only on the first n − x = ( x , . . . x n ) then d ( φ k ( x ) , e kR x ) = d ( A k · e kR ( x ) , e kR x ) = d ( A k x, x )We can apply a further isometry such that x, A k x both lie on the plane x n = 1and by Lemma 4.6 it is still true that k x k x n ≤ e D and hence if we let π be theprojection map onto the first n − e D ≥ k x k = k π ( x ) k + 1 and so p e D − ≥ k π ( x ) k . In particular, if we restrict to the hyperplane R n − × { } the a -ball around eachof the translates A k x lies within the 2 e D -ball in R n − ×
1. Indeed, let x ′ lie in suchan a -ball, then as a < k π ( x ′ ) k ≤ k π ( x ) k + a k π ( x ) k + a < e D − a p e D − a < e D Now let vol ( B ( r )) denote the volume of the r -ball in Euclidean ( n − f ( a ) = vol ( B (2 e D )) vol ( B ( a )) . If k > f ( a ) then there must be some pair of points A i x, A j x with i < j suchthat their a -balls overlap, otherwise all the balls are disjoint but the sum of theirvolumes is greater than the volume of the 2 e D -ball containing them. Thus the result follows as vol ( B (2 e D )) vol ( B ( a )) = (2 e D ) n − (cid:0) a (cid:1) n − vol ( B (1)) vol ( B (1)) = (cid:18) (4 e D ) a (cid:19) n − and if the neighbouring balls of A i x, A j x overlap then so do the balls round x, A j − i x and so A j − i x is within Euclidean (and hence also hyperbolic) distance a of x asrequired. (cid:3) Theorem 4.8.
Suppose that M is a finite volume hyperbolic n -manifold ( n ≥ with systole(s) of length R ≤ ǫ n , where ǫ n is the Margulis constant in dimension n . Then the distance from a systole to the boundary of the Margulis tube containingit is bounded below by n log (cid:18) R (cid:19) + log ( ǫ n ) − log (4) . Proof.
Let γ be a systole of M . We consider N = N ( γ, D ) in an attempt to boundbelow the maximum value of D such that this neighbourhood lies in the thin partof M .By definition γ is a geodesic and so the corresponding isometry of hyperbolicspace in the deck group of M is a loxodromic with geodesic axis ˜ γ which projectsdown to γ under the covering map. By conjugation this can be modelled in theupper half space model of hyperbolic space by the isometry φ : v A · e R v with geodesic axis the geodesic from 0 to ∞ where R is the translation length of φ and A is an orthogonal map on the first n − x := ( x , . . . , x n ) be a point in H within distance D from the axis of φ , so in particular k x k x n < cosh ( D ) < e D and let x ′ := ( x , . . . , x n − , e kR x n ) be thevertical translate of x by hyperbolic distance kR . Then d ( x, φ k ( x )) ≤ d ( x, x ′ ) + d ( x ′ , e kR x ) + d ( e kR x, φ k ( x )) . Now d ( x, x ′ ) is just the length of the vertical geodesic from x to x ′ . This is log ( e kR x n ) − log ( x n ) = log ( e k R ) = kR .Furthermore, the distance from x ′ to e kR x is bounded above by their distancein the horosphere, which is Euclidean up to scaling. So if π ( x ) represents theprojection to the first n − d ( x ′ , e kR ( x )) ≤ e kR x n k e kR π ( x ) − π ( x ) k = ( e kR − k π ( x ) k e kR x n ≤ ( e kR − e D )and so by Lemma 4.7 d ( x, φ ( x )) ≤ kR + ( e kR − e D + a. At this point we note that the following statement follows by simple rearrangmentand taking logarithms(1)
D < n log (cid:18) R (cid:19) + log ( ǫ n ) − log (4) ⇔ R (cid:18) e D ǫ n (cid:19) n < YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 15 Thus, if we set a = ǫ n and require D < n log (cid:18) R (cid:19) + log ( ǫ n ) − log (4)then for k < (cid:16) e D a (cid:17) n − it is both true that kR = R (cid:18) e D ǫ n (cid:19) n − < R (cid:18) e D ǫ n (cid:19) n < kR (2 e D + 1) < (cid:18) e D ǫ n (cid:19) n − R (4 e D ) < R (cid:18) e D ǫ n (cid:19) n ǫ n < ǫ n Combining this with the fact that e x − < x on the interval (0 ,
1) gives that d ( x, φ ( x )) ≤ kR + ( e kR − e D + a ≤ (1 + 2 e D ) kR + ǫ n < ǫ n and hence if x is within distance D of the axis of φ then x maps to the thin part of M . Thus the thick-thin decomposition implies that the diameter of the Margulistube is at least 1 n log (cid:18) R (cid:19) + log ( ǫ n ) − log (4) (cid:3) If our desire is to give an explicit relation bound on injectivity radius and di-ameter as a function of dimension, and number of n -simplices, then we need tounderstand ǫ n as a function of n . Fortunately, this an area of much research andlower bounds are known. Robert Meyerhoff gives a bound on ǫ [Mey87]. Forgeneral n , a result of Ruth Kellerhals [Kel04] gives a lower bound for ǫ n . Theorem 4.9 (Meyerhoff [Mey87]) . With the definition of the thick-thin decompo-sition given in Theorem 4.4, the value of ǫ is at least . . Theorem 4.10 (Kellerhalls [Kel04]) . The value of ǫ n is bounded below by π ) n . In fact Kellerhals’ paper gives a tighter lower bound than this, but we shall usethis coarser bound for simplicity of expression.
Theorem 4.11.
Given a closed hyperbolic n -manifold ( n ≥ triangulated by tn -simplices, let R be the length of a systole in M , then R ≥ ( nt ) O ( n t ) . Proof.
Let X be the hyperbolic manifold defined in Lemma 3.9, then by Lemma3.17 we know that X admits a cover by t immersed hyperbolic n -simplices withedge lengths bounded above by some bound B ( t ) with B ( t ) ≤ ( nt ) O ( n t ) . Pick some ǫ > π ) n so that M (0 ,ǫ ) satisfies the conclusion of Theorem 4.4. This ispossible as the bound in Theorem 4.10 is not strict.If the thin part M (0 ,ǫ ) is empty, then R ≥ π ) n and so the conclusion holds.Suppose the thin part is non empty, then Theorem 4.8 applies. Consider a systole γ . This curve has length R and lies in the centre of one of the tubes of the thinpart. We know from the above that the diameter of this tube is at least1 n log (1 /R ) + log ( ǫ n . However, we know that M is covered by t n -simplices of bounded edge length,and thus of bounded diameter, and so the diameter of M is at most tB ( t ) ≤ t ( nt ) O ( n t ) and hence 1 n log (1 /R ) + log ( ǫ n ≤ t ( nt ) O ( n t ) . Simplifying big O notation further and applying Theorem 4.10 log (cid:18) R (cid:19) ≤ ( nt ) O ( n t ) + nlog (4(6 π ) n ) . Taking exponentials on both sides gives1 R ≤ e ( nt ) O ( n t ) (24 π ) n which simplifies, after rearrangement, to R ≥ ( nt ) O ( n t ) (cid:3) The Cusped Case
With some small but technical adaptations, the arguments from the closed casehold in the cusped (finite volume) case where M is given by a finite semi-idealtriangulation and M admits a finite volume hyperbolic structure. Lemma 5.1.
The homotopy equivalence defined in Lemma 3.9 is surjective also inthe finite-volume case.Proof.
In the proof of Lemma 3.9 we showed that the homotopy equivalence F ishomotopic to a homeomorphism F ′ , and in fact the homotopy fixes ideal vertices,so the entire homotopy extends to the space we get when we add the ideal verticesback in (replace all ideal simplices with non-ideal ones). Call this pseudomanifold M ′ and the target X ′ . Then the n th homology of M ′ , X ′ is Z with fundamentalclass [ M ′ ] given by the triangulation formed by replacing the ideal vertices withconcrete ones. Now note that [ M ′ ] generates both H n ( M ′ ) and H n ( M ′ , M ′ − x ) for x ∈ M , and so, as F [ M ′ ] is homotopic and thus homologous to F ′ [ M ′ ], we knowthe same is true for F [ M ′ ] in H n ( X ′ ) and H n ( X ′ , X ′ − x ) for x ∈ X ′ . Thus, thesame argument from Lemma 3.10 holds here and F is a surjection on M ′ and thusalso on M . (cid:3) In order to use Calabi-Weil in the finite volume case we need control over para-bolic isometries which can be tricky, especially in higher dimensions where parabolicsubgroups are not necessarily translation groups. We solve this problem in higherdimensions by using a stronger statement of rigidity, due to Garland and Raghu-nathan instead.
YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 17 Theorem 5.2 (Theorem 7.2 in [GR70]) . Let G be a lattice in Isom ( H n ) for n ≥ .The identity representation ρ : G → Isom ( H n ) := Λ lies in a connected componentof Hom ( G ; Λ) made up entirely of conjugates of ρ . Lemma 5.3.
Let M be a triangulated finite volume hyperbolic n -manifold, n ≥ . There is a system of polynomial inequalities as in Theorem 2.1 such that thefollowing all hold: • The solution set consists of cocycles in C ( M, SO ( n, • The solution set has a component consisting entirely of cocycles which in-duce faithful lattice representations. • The system consists of < ( n + 1) t polynomials in < ( n + 1) t variables,with degree bounded by ( n + 1) t and all coefficients ± .Furthermore, this systems contains variables for the lengths of all non-ideal edges.Proof. The statements and proofs of Lemmas 3.15 and 3.16 hold as is in the cuspedcase (for n ≥
4) if we replace mention of edges and simplices by non-ideal edgesand non-ideal simplices. The only other difference being that instead of applyingCalabi-Weil Rigidity, we apply Theorem 5.2. (cid:3)
In the 3-dimensional case, Garland and Raghunathan’s result no longer holds,and Calabi-Weil rigidity require that we have control over where parabolics aresent. For this we consider
Isom + ( H ) ∼ = P SL (2 , C ) instead, while sticking withthe hyperboloid model, using an action described in the proof. The reason behindthis is that P SL (2 , C ) offers a remarkably easy method of verifying parabolicity,namely, a matrix in P SL (2 , C ) is parabolic iff its trace is ±
2, or equivalently, itstrace squares to 4.There is one small hitch with looking at representations into
P SL (2 , C ). To applythe results of section 2 we need to be looking at elements of a variety. Fortunately,the following tells us we can equivalently look at representations into SL (2 , C ). Theorem 5.4 (Corollary 2.3 [Cul86] ) . If a discrete subgroup G of P SL (2 , C ) hasno 2-torsion then it lifts to SL (2 , C ) . That is there is a homomorphism G → SL (2 , C ) such that the composition with the natural projection SL (2 , C ) → P SL (2 , C ) is the identity on G . Combining this with the fact that connected components of
Hom ( G, SL (2 , C ))map down to connected components of Hom ( G, P SL (2 , C )) tells us that if we applythe results of Section 2 to a system such as those created so far but with imagesin SL (2 , C ), then we can find a connected component of the solution set whichcontains a solution inducing a faithful lattice representation, and thus by Theorem5.2 the whole component consists of such representations.Because of the focus on parabolic elements, which correspond to loops whichare freely homotopic to ideal vertices, it will be useful to be able to talk aboutthe parabolics corresponding to each individual ideal vertex with the followingdefinition. Definition 5.5 (Cusp Fundamental Group) . We define the fundamental group ofa given cusp in M to be fundamental group of the star of a chosen ideal vertex. Note that choosing a path from our chosen basepoint for M to our chosen basepointfor the star of an ideal vertex defines an embedding of the cusp fundamental groupinto the fundamental group of M . All such choices define conjugate embeddings.We can now develop a system of polynomials which suits the final case of cuspedhyperbolic 3-manifolds. Lemma 5.6.
Let M be a finite volume hyperbolic -manifold triangulated by t tetrahedra with fundamental group G . There is a system of polynomial inequalitiesas in Theorem 2.1 such that the following all hold: • The solution set consists of cocycles in C ( M, SL (2 , C )) which induce rep-resentations in Hom par ( G, SL (2 , C )) . • The solution set has a component consisting entirely of cocycles each ofwhich induces a faithful lattice representation. • The system has each of its measures of complexity
N, κ, d, M (as in Section2) bounded above by a constant multiple of t .Furthermore, this systems contains variables for the lengths of all non-ideal edgesin the image of the homotopy equivalence defined in Lemma 3.9.Proof. Outside of the handling of parabolics, the proofs here are very similar to the SO ( n,
1) case, so we will only outline where the proofs differ from those given forLemmas 3.15 and 3.16.To get a system of polynomials which has cocycles in C ( M, SL (2 , C )) as itssolutions (not yet including variables for edge lengths and not necessarily inducingrepresentations in Hom par ( G, SL (2 , C )), the same proof as for SO ( n,
1) holds. Weget some constant multiple of t as a bound for the number of polynomials andvariables, and the polynomials have degree at most quadratic with coefficients ± t and the degree and coefficientcomplexity of the polynomials are both bounded by a constant. This uses thefollowing action of SL (2 , C ) on the hyperboloid model.One can represent the point ( x, y, z, t ) in the hyperboloid model by the matrix X = (cid:18) t + z x − iyx + iy t − z (cid:19) and then A ∈ SL (2 , C ) acts by X AXA ∗ . Note that taking determinant gives the standard quadratic form used to define thehyperboloid and the action preserves determinant.Finally the big difference in the cusped case is that we need to check that par-abolic subgroups of π ( M ) are indeed sent to parabolics in P SL (2 , C ) for eachrepresentation induced by a cocycle solution to our system. In three dimensionsit is enough to check that generators of the cusp fundamental group are sent toparabolics which fix the same point at infinity, as parabolics in Isom + ( H ) re-strict to translations on horospheres about their fixed point, and so products ofsuch parabolics are also parabolics, as non-trivial products of translations are non-trivial translations. Note this is not true in higher dimensions where parabolics YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 19 don’t have to restrict to translations and so two parabolics with the same fixedpoint at infinity can have a non-trivial elliptic as their product.To do this we take generating sets of each cusp fundamental group as describedin the proof of Lemma 3.9. That is we pick a spanning tree of the link of eachideal vertex, then the remaining edges define loops in the link, each of which canbe homotoped arbitrarily close to the ideal vertex and thus maps to a parabolic,and which together generate the fundamental group of the cusp. As there are lessthan 6 t total edges, we see that there are less than 6 t such generating curves, eachof which is a product of less than 6 t edges. So for each curve we define polynomialswhich require that the square of the trace of the matrix associated to each suchcurve be 4. As this matrix is a product, the degree of these polynomials could beas much as 6 t . We also introduce new variables which correspond to the sharedfixed points of the generating set of a cusp, and polynomials requiring that they befixed.At all times however, all measures of complexity of this system are bounded bya multiple of t .Thus we have a system of polynomials whose solutions correspond to cocyclesin C ( M, SL (2 , C )) and also to the lengths of edges in the image of the homotopyequivalence. We also know that parabolic generators (and hence the entirety ofeach parabolic subgroup) are mapped to parabolic elements in the induced repre-sentation into P SL (2 , C ). We know that there is a solution to this system inducinga faithful lattice representation and the entire connected component of the solutionset containing this solution induces representations into P SL (2 , C ) which all liein the same component of Hom par ( π ( M ) , P SL (2 , C )) and hence, by Calabi-WeilRigidity these representations are all themselves faithful lattices (cid:3) As in the closed case, the following lemma follows by applying Theorem 2.1 tothe systems of polynomials defined in Lemmas 5.3 and 5.6 and simplifying big Onotation.
Lemma 5.7.
Let M be a finite volume hyperbolic manifold equipped with a semi-ideal triangulation by t semi-ideal n -simplices. Then there exists a homotopy equiv-alence F as described in Lemma 3.9 such that the image of each non-ideal edge haslength l ( e ) bounded in the following way: l ( e ) ≤ ( nt ) O ( n t ) . Theorem 5.8.
Given a finite volume hyperbolic n -manifold ( n ≥ triangulatedby t semi-ideal n -simplices, let R be the length of a systole in M , then R ≥ ( nt ) O ( n t ) . Proof.
Throughout, we let B ( t ) be a bound on the edge length of the image ofedges under the homotopy equivalence F . Let ¯ X be the image in X under F of thenon-ideal simplices in M . For each simplex in M , the diameter of the image of itsnon-ideal part is bounded by the maximum length of its edges, hence by B ( t ), andthus the diameter of ¯ X is bounded by tB ( t ). Note also that, as the links of idealvertices consist entirely of non-ideal simplices, the images of links of ideal verticeslie in ¯ X .We aim to show that points in X either lie in a bounded neighbourhood of ¯ X ,or they lie in the subset of the thin part made up of cusp neighbourhoods. Note that the complement of ¯ X is covered by the images of the stars of the idealvertices of M . Consider one such vertex v i , and its star st ( v i ), let S i denote theimage of st ( v i ) in X and L i denote the image of lk ( v i ). Then S i is a union ofgeodesic semi-ideal n -simplices. Take a collection of lifts (one per simplex) of thesesemi-ideal n -simplices in H n (the upper half space model) such that the collectionis connected, and shares the same point on the boundary. We can perform anisometry of H n taking this shared point to the point at infinity. We shall call thiscollection T i , and note that we still have a canonical map from T i into X .Consider the map φ d which takes each point p in the upper half space model andtranslates it a hyperbolic distance d along the unique (vertical) geodesic through p and the point at infinity. Then this map scales distances in H n by1 e d . Let ψ d be the map on S i which is induced by φ d | T i , then any edge L i has itslength scaled by the scale factor above. Now as L i ⊆ ¯ X , its diameter is boundedby tB ( t ). Thus setting d = log tB ( t ) ǫ n , we know that for all d ≥ d , the image of L i under ψ d has diameter bounded abovein the following way Diam ( ψ d ( L i )) < e log t ( B ( t )) ǫ t ( B ( t )) = ǫ n Thus as the union of all ψ d ( L i ) covers S i , every point p lies in some ψ d ( L i ). If d ≤ d , then p lies in a d neighbourhood of ¯ X . If d > d , then there is some curveof length < ǫ n based at p which lies entirely in ψ d ( L i ), and hence p , and in factall of ψ d ( S i ) lies in the thin part of X . As ψ d ( S i ) is a neighbourhood of an idealvertex, it must lie in one of the cusp neighbourhoods of the thin part.The thin part of a manifold decomposes into cusp neighbourhoods and tubeswhich are disjoint. Furthermore, there are no geodesics in the cusp neighbourhoodsas translation towards the end of the cusp decreases the length of a curve. Thuseither all systoles lie in the thick part and have length bounded below by 2 ǫ n orthey lie in the thin part and are disjoint from all the cusp neighbourhoods.Each systole must then lie in the d -neighbourhood of ¯ X . This means that theentire 1-skeleton of ¯ X lies within distance diam ( ¯ X ) + d < tB ( t ) + log tB ( t ) ǫ n of each systole. By using Theorem 5.7 as a bound for B ( t ), using Theorem 4.10 asa bound for ǫ n and simplifying big O notation, we get that ¯ X lies within distance( nt ) O ( n t ) . of the systole. This is therefore also an upper bound for the distance from eachsystole to the boundary of the tube containing it. Indeed if ¯ X lies in this tube,then a generating set for π ( X ) lies in this subspace which has fundamental group Z , and so π ( X ) is a subgroup of Z , a contradiction as π ( X ) is the fundamentalgroup of a finite volume hyperbolic manifold and so cannot be Z . YSTOLE LENGTH IN HYPERBOLIC n -MANIFOLDS 21 Thus by Theorem 4.81 n log (cid:18) R (cid:19) + log ( ǫ n ) − log (4) < ( nt ) O ( n t ) and hence we can rearrange this (simplifying big O notation) to log (cid:18) R (cid:19) ≤ ( nt ) O ( n t ) + nlog ( n (6 π ) n )and thus simplifying again R ≥ ( nt ) O ( n t ) (cid:3) References [AR00] Colin Adams, Alan Reid Systoles of Hyperbolic 3-Manifolds
Mathematical Proceedingsof the Cambridge Philosophical Society (1), (2000), pp.103-110 .[Ago06] Ian Agol. Systoles of Hyperbolic 4-Manifolds Preprint, arXiv:math/0612290.[BT11] M. Belolipetsky, S. Thomson. Systoles of Hyperbolic Manifolds.
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