A bound for diameter of arithmetic hyperbolic orbifolds
aa r X i v : . [ m a t h . M G ] M a r A BOUND FOR DIAMETER OF ARITHMETIC HYPERBOLICORBIFOLDS
MIKHAIL BELOLIPETSKY
Abstract.
Let O be a closed n -dimensional arithmetic (real or complex) hyper-bolic orbifold. We show that the diameter of O is bounded above by c log vol( O ) + c h ( O ) , where h ( O ) is the Cheeger constant of O , vol( O ) is its volume, and constants c , c depend only on n . Introduction
Let V be a closed real or complex hyperbolic manifold of dimension n ≥
4. In[BS87], Burger and Schroeder proved that diam( V ) ≤ λ ( V ) ( β n log vol( V )+ α n ), where λ ( V ) is the first non-zero eigenvalue of the Laplacian on V and the constants α n , β n depend only on n . It is well known that this inequality does not hold for hyperbolic2 and 3–manifolds (see Remark (v) in [BS87]). In this article we prove that aninequality of this type still holds if we restrict to arithmetic O can have a small injectivity radius only if its volume is very large. After thiswe use an argument due to Brooks [Br92], which we extend here to orbifolds. Thelatter extension again requires arithmeticity assumption, which allows us to boundthe order of singularities in terms of vol( O ). Dealing with volumes and areas in thepresence of singularities requires some results from geometric measure theory. Themain theorem of the paper is stated as follows: Theorem 1.
Let O be a closed arithmetic (real or complex) hyperbolic orbifold ofdimension n ≥ . Then the diameter diam( O ) , the Cheeger constant h ( O ) and thevolume vol( O ) satisfy the inequality diam( O ) ≤ c log vol( O ) + c h ( O ) Belolipetsky is partially supported by CNPq and FAPERJ research grants. with constants c , c depending only on n . In dimensions 2 and 3 this result is new for orbifolds and for manifolds. In di-mensions n ≥ h ( O ) in the denominator follows from [BS87, Theorem 2] and Cheeger’s inequality λ ( O ) ≥ h ( O ) . As an example of a real hyperbolic n –orbifold one can consider asphere S n with a sufficiently complicated singular set. For example, a figure–8 knotin S with singularity of order 2 gives an arithmetic 3–orbifold. It is not known ifarithmetic hyperbolic orbifolds with underlying space S n exist for large n .An interesting class of spaces to which Theorem 1 applies are the congruence arith-metic hyperbolic orbifolds . These, in particular, include arithmetic orbifolds whosegroups are maximal discrete subgroups. The first non-zero eigenvalue of the Lapla-cian for these orbifolds is uniformly bounded below by a constant which depends onlyon the dimension (see [BS91] for the real hyperbolic case and [Cl03] in general). ByBuser’s inequality (generalized to orbifolds) it implies a uniform lower bound for theCheeger constant [Bus82]. Hence in this case we will have an inequalitydiam( O ) ≤ c log vol( O ) + c . The paper is organized as follows. In Section 2 we collect various results aboutmetric properties of arithmetic orbifolds. In Section 3 we prove bounds for injectivityradius and volume of small balls in arithmetic hyperbolic orbifolds and finish withthe proof of Theorem 1. 2.
Preliminaries
Hyperbolic orbifolds.
Let Γ be a group of isometries of a Riemannian manifold X acting properly discontinuously. If Γ is torsion-free, then the quotient space X/ Γhas a structure of a Riemannian manifold. More generally, the group Γ may have finitepoint-stabilizers, and X/ Γ is endowed a structure of a (good) Riemannian orbifold .An orbifold has an atlas of maps locally identifying it with a quotient of an open setin X by a finite group of isometries. The term orbifold was coined by W. Thurstonin late 1970’s. A similar concept was introduced by Satake in [Sat56], where he usedthe term V-manifold .In this paper X will be always a real or complex hyperbolic space: X = H n R or H n C .2.2. Geodesics and diameter.
Metric properties of Riemannian orbifolds werestudied by Borzellino in his thesis [Bor92]. In particular, he showed that there isa natural distance d on O with which ( O , d ) becomes a length space, and if ( O , d ) iscomplete, any two points can be joined by a minimal geodesic realizing the distancebetween them [Bor92, Theorem 40, p. 21]. The singular set Σ of a good Riemann-ian orbifold O is locally convex [Bor92, Proposition 1, p. 31]. Moreover, a geodesicsegment cannot pass through Σ unless it starts and/or ends there (see [Bor92, Theo-rem 3, p. 32]). A consequence of this fact is that the complement of Σ in O is convexas all points in O \
Σ can be joined by some segment, and Σ cannot disconnect O . BOUND FOR DIAMETER OF ARITHMETIC HYPERBOLIC ORBIFOLDS 3
We define the diameter of O as the supremum of the distances between points in O .For a closed O the supremum is achieved and there is a geodesic joining x, y ∈ O whose length is equal to diam( O ).2.3. Injectivity radius.
Let O = X/ Γ be a closed hyperbolic orbifold and let π : X → O be the covering map. The elements of the group Γ fall into two types: elliptic are those which have fixed points in X and hyperbolic are those which actfreely. For a hyperbolic isometry γ ∈ Γ its displacement at x ∈ X is defined by ℓ ( γ, x ) = d ( x, γx ) and the displacement of γ (also called its translation length ) is ℓ ( γ ) = inf x ∈ X ℓ ( γ, x ) . It is equal to the displacement of γ at the points of its axis . We will define theorbifold injectivity radius by r inj ( O ) = inf { ℓ ( γ ) } , where the infimum is taken overall hyperbolic elements γ ∈ Γ. It is equal to half of the smallest length of a closedgeodesic in O . When O is a manifold, this definition is equivalent to the usualdefinition of the injectivity radius as the supremum of r such that any point p ∈ O admits an embedded ball B( p, r ) ⊂ O . This is not the case in general; the points inthe singular set only admit embedded folded balls (cf. [Sam13]).2.4. Cheeger’s constant.
We define the
Cheeger constant h ( O ) of a closed orbifold O by h ( O ) := inf (cid:16) area( ∂A )min { vol( A ) , vol( O \ A ) } (cid:17) , where A ⊆ X is an open subset with Hausdorff measurable boundary ∂A .By Toponogov’s theorem for orbifolds [Bor92, Theorem 1, p. 28], a hyperbolic n –orbifold is an Alexandrov space with curvature bounded from below. In [OS94], Otsuand Shioya proved that the singular set Σ has Hausdorff dimension ≤ n −
1. Moreover,they showed that there exists a C –Riemannian structure on O \ Σ ⊂ O satisfyingthe following:(1) There exists O ⊂ O \ Σ such that
O \ O is of n -dimensional Hausdorffmeasure zero and that the Riemannian structure is C / –continuous on O .(2) The metric structure on O \
Σ induced from the Riemannian structure coin-cides with the original metric of O .It follows that we can also compute the volumes in the definition of Cheeger’s constantusing the Otsu–Shioya metric. Concerning the area of the boundaries, when O isorientable the singular set has codimension 2 and we can again use the metric. When O is non-orientable the singular hypersubsets belong to the boundary of O and so donot enter into the formula. We will not use these facts in the proof of the theorem butthey are important for understanding the metric structure of hyperbolic orbifolds. MIKHAIL BELOLIPETSKY
Arithmetic orbifolds.
Let H be a linear semisimple Lie group with trivialcenter, in our case we have H = Isom( X ). Let G be an algebraic group defined over anumber field k such that G( k ⊗ Q R ) is isogenous to H × K , where K is a compact Liegroup. Consider a natural projection φ : G( k ⊗ Q R ) → H . The image of the group of k -integral points G( O k ) and all subgroups Γ < H which are commensurable with it arecalled arithmetic subgroups of H defined over k . Borel and Harish–Chandra provedthat arithmetic subgroups are lattices, i.e., they are discrete and have finite covolumein H [BHC62]. Their quotient spaces O = X/ Γ are called arithmetic orbifolds , theycome together with an associated number field k , the field of definition . Number–theoretical properties of the field k can be used to extract important informationabout geometry of O . This principle underlines the result of the present paper.We refer to [WM15] for a comprehensive introduction to the theory of arithmeticsubgroups and their quotient spaces.3. Proof of Theorem 1
We first prove two lemmas. Similar results for real hyperbolic orbifolds were pre-viously proved in [AB19, Section 3] (see also [Bel10]). Here we extend them to thecomplex hyperbolic case. We repeat some details for the readers convenience.
Lemma 3.1 (A bound for injectivity radius) . Given a closed n -dimensional arith-metic hyperbolic orbifold O = X/ Γ of sufficiently large volume, we have r inj ( O ) ≥ a (cid:18) log log log vol( O ) b log log vol( O ) b (cid:19) with the constants a , b > depending only on X .Proof. Let γ ∈ Γ be a hyperbolic transformation. The eigenvalues of γ considered asan element of SL( n + 1 , C ) are e ± c ℓ ( γ ) and n − c = c ( X ) (for the real hyperbolic case it is by [Gr62, Proposition 1(1,4)], andfor the complex hyperbolic case it follows from [Gol99, Section 3.3.3]).We would like to relate e c ℓ ( γ ) to the Mahler measure of a certain polynomial natu-rally associated to γ . To this end we can adapt the argument of [Gel04, Section 10].Let H ◦ be the identity component of the group H = Isom( X ). It is center-free andconnected so we can identify it with its adjoint group Ad( H ◦ ) ≤ GL( g ), where g denotes the Lie algebra of H . We have Γ ′ = Γ ∩ H ◦ , a cocompact arithmetic lattice,and γ ∈ Γ ′ . Since Γ ′ is arithmetic, there is a compact extension H ◦ × K of H ◦ and a Q –rational structure on the Lie algebra g × k of H ◦ × K , such that Γ is theprojection to H ◦ of a lattice ˜Γ, which is contained in ( H ◦ × K ) Q and commensurableto the group of integral points ( H ◦ × K ) Z with respect to some Q –base of ( g × k ) Q .By changing the Q –base we can assume that ˜Γ is contained in ( H ◦ × K ) Z . Thus thecharacteristic polynomial P ˜ γ of any ˜ γ ∈ ˜Γ is a monic integral polynomial of degreeat most m deg( k ), where k is the field of definition of the arithmetic group and the BOUND FOR DIAMETER OF ARITHMETIC HYPERBOLIC ORBIFOLDS 5 dimension bound m depends only on the type of H . Since K is compact, any eigen-value of ˜ γ with absolute value different from 1 is also an eigenvalue of its projectionto H ◦ . Therefore, e c ℓ ( γ ) = M ( P ˜ γ ) , where the Mahler measure of an integral monic polynomial P ( x ) of degree d is definedby M ( P ) = d Y i =1 max(1 , | θ i | ) , where θ ,. . . , θ d are the roots of P ( x ).Hence we have ℓ ( γ ) ≥ c log M ( P ˜ γ ) . This implies that r inj ( O ) ≥ min { c log M ( P ˜ γ ) } , where the minimum is taken overall ˜ γ ∈ ˜Γ which project to hyperbolic elements in Γ ′ . Moreover, our argument showsthat the degrees of the irreducible integral monic polynomials whose Mahler measuresappear in this bound satisfy d ≤ m deg( k ) . By Lehmer’s conjecture the Mahler measures of non-cyclotomic polynomials areexpected to be uniformly bounded away from 1. A special case of this conjecture alsoknown as the Margulis conjecture implies a uniform lower bound for the lengths ofclosed geodesics of arithmetic locally symmetric n –dimensional manifolds (see [Gel04,Section 10]). These conjectures have attracted a lot of interest but still remain wideopen. For our estimate we will take advantage of the known quantitative number-theoretical results towards Lehmer’s problem.In [Dob79], Dobrowolski proved the following lower bound for the Mahler measure:(1) log M ( P ) ≥ c (cid:18) log log d log d (cid:19) , where d is the degree of the polynomial P and c > d to the volume by using an important inequality relatingthe volume of a closed arithmetic orbifold and the degree of its field of definition:(2) deg( k ) < c log vol( O ) + c . For hyperbolic orbifolds of real dimension n ≥ x the function log xx is monotonically decreasing, hence forsufficiently large volume we obtain the inequality of the lemma with a = c c and b = c . (cid:3) MIKHAIL BELOLIPETSKY
The points of orbifold at which the injectivity radius is small form a thin part .The Margulis lemma implies that hyperbolic orbifolds also have thick part where theinjectivity radius is bigger than the Margulis constant. We will use Margulis’ lemmain the form given in [Sam13, Section 2.2]. Let µ n and m n be the constants definedthere which depend only on the space X , and let ε = min { µ n , r inj ( O )4 m n } . Lemma 3.2 (A bound for the volumes of balls) . A ball of radius r ≤ ε in an arith-metic hyperbolic orbifold O = X/ Γ has volume vol(B( r, x )) ≥ v r a log vol( O ) + b , where v r denotes the volume of a ball of radius r in X and the constants a , b > depend only on X .Proof. We first bound the order q of finite subgroups F <
Γ in terms of volume. Wedo this by applying Margulis lemma to the discrete subgroups of O( n ) or U( n ) as itwas done in [ABSW08] in the case of orthogonal groups. Consider a k –embeddingof Γ into GL( m, k ) with m = n + 1 if X = H n C or H n R with n even, m = 2( n + 1)if X = H n R with n is odd and = 7, and m = 24 if X = H R . The existence of suchembedding follows from the classification of Q –forms of the orthogonal and unitarygroups (see [WM15, Section 18.5]).Let A ∈ GL( m, k ) be a torsion element of order t ≥
2. Let λ , . . . , λ m be theeigenvalues of A . As A is a matrix over k , its eigenvalues split into groups of conjugatesunder the action of Gal(¯ k/k ). The equality A t = Id implies that the eigenvalues areroots of unity. Let t , . . . , t l be their orders, so t = lcm( t , . . . , t l ). If λ is an eigenvaluethen all its Gal(¯ k/k )-conjugates are also eigenvalues of A , which implies φ k ( t ) + · · · + φ k ( t l ) ≤ m, where φ k ( t ) denotes a generalized Euler φ -function defined as the degree over k of thecyclotomic extension k ( µ t ), µ t is a primitive t –root of unity.It is clear that the following inequalities are satisfied for φ k ( t ) and the Euler φ -function: φ ( t ) / deg( k ) ≤ φ k ( t ) ≤ φ ( t ) . We have φ ( t i ) ≤ m deg( k ) , which implies φ ( t ) ≤ φ ( t ) · · · φ ( t l ) ≤ ( m deg( k )) m . By using the well known inequality φ ( t ) ≥ √ t/
2, we obtain(3) t ≤ m deg( k )) m ≤ c deg( k ) m , where the constant c depends only on m .This is a bound for the order of finite cyclic subgroups of Γ, which can be used togive an upper bound for q . By the Margulis lemma, the constant m n has a property BOUND FOR DIAMETER OF ARITHMETIC HYPERBOLIC ORBIFOLDS 7 that if
F < O( n ) or U( n ) is a finite subgroup, then there is an abelian subgroup A < F such that [ F : A ] ≤ m n (see [Sam13, Theorem 2.1]). We may find commoncomplex eigenspaces of the elements of A : U , . . . , U k , and real eigenspaces V , . . . , V l ,where k/ l ≤ m , such that A acts on U i as a cyclic group A i , and A acts on V i as ±
1. We may embed A in Q ki =1 A i × ( Z / Z ) l , acting on Q ki =1 U × Q lj =1 V j .Thus | A | ≤ l Q ki =1 | A i | , and by the previous argument, | A i | ≤ c deg( k ) m , since agenerator of A i is the projection of an element of A . Thus we obtain(4) q ≤ m n | A | ≤ c deg( k ) c , with the constants c , c > m .Together with (2) it implies(5) q ≤ c log vol( O ) c + c , with the constants depending only on m , and hence only on X .We now use the Margulis lemma once again to show that every ε –ball in X mapsto O with multiplicity at most q at each point. The constants µ n and m n in theMargulis lemma have the following property. For any x ∈ X and any t ∈ R , let Γ t ( x )denote the subgroup of Γ generated by the elements that move x by distance less than t . Then if t ≤ µ n and if Γ t ( x ) is infinite, there is an element in Γ of infinite order thatmoves x by distance less than 2 m n t [Sam13, Lemma 2.3]. We have chosen ε such that2 ε ≤ µ n and 2 m n (2 ε ) ≤ r inj ( O ). By definition, every element in Γ of displacementless than r inj ( O ) has a fixed point and therefore has finite order. Thus, Γ ε ( x ) mustbe finite for all x ∈ X . Let x , . . . , x k be points in some ε –ball in X that all map tothe same point of O . Then they are all in the orbit of x under Γ ε ( x ), and Γ ε ( x )has order at most q satisfying (5) as we have shown this for every finite subgroupof Γ. Hence k ≤ q and the multiplicity of the map is at most q on the ε -ball. Thisfinishes the proof of the lemma. (cid:3) Proof of the theorem.
By the theorems of Wang and Borel [Wa72, Bor81] there arefinitely many arithmetic hyperbolic n -orbifolds of bounded volume. Hence we canassume that vol( O ) is sufficiently large and compensate the remaining ones by theadditive constant.We now adapt the argument of [Br92].Let r := a (cid:16) log log log vol( O ) b log log vol( O ) b (cid:17) be the bound for injectivity radius from Lemma 3.1.As vol( O ) is large, we have r < ε from Lemma 3.2. Pick a point x in O and considera ball B( r, x ) of radius r about x . If r ≤ r , then the volume of the ball is estimatedby Lemma 3.2. We now want to estimate the volume V ( r ) = vol(B( r, x )) for r > r .By the coarea formula (cf. [Fed69, Section 3.2]) and the definition of the Cheegerconstant we have V ′ ( r ) V ( r ) ≥ h ( O ) , MIKHAIL BELOLIPETSKY as long as V ( r ) < vol( O ).By integrating we obtain V ( r ) ≥ e h ( O )( r − r ) V ( r ) until V ( r ) = 12 vol( O ) . This will happen when r = r = 1 h ( O ) (cid:16) log vol( O ) − log(2) − log V ( r ) (cid:17) + r . Therefore, for any x, y ∈ O we have B( r , x ) ∩ B( r , y ) = ∅ , and hence diam( O ) ≤ r .By Lemma 3.2 and the value of r provided by Lemma 3.1 we conclude that the terms − log V ( r ) and r are much smaller than log vol( O ). (cid:3) References [ABSW08] I. Agol, M. Belolipetsky, P. Storm, and K. Whyte, Finiteness of arithmetic hyperbolicreflection groups,
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