On volumes of hyperbolic orbifolds
In this paper we use H. C. Wang’s bound on the radius of a ball embedded in the fundamental domain of a lattice of a semisimple Lie group to construct an explicit lower bound for the volume of a hyperbolic –orbifold.
Let denote hyperbolic –space; the unique simply connected –dimensional Riemannian manifold of constant sectional curvature . A hyperbolic –orbifold is a quotient , where represents a discrete group of orientation-preserving isometries. A hyperbolic –orbifold is a manifold when contain no elements of finite order. In , Martin constructed a lower bound for , the largest number such that every hyperbolic –manifold contains a round ball of that radius; see also Friedland and Hersonsky . From this one can compute, in each dimension, an explicit lower bound for the volume of a hyperbolic –manifold.
The purpose of this paper is to give an explicit lower bound for the volume of a hyperbolic –orbifold, again depending only on dimension. The result of this article is more general than what was achieved in the prequel . Our work also significantly improves upon the volume bounds of  and , even though we consider a larger category of orbit spaces.
We define a Riemannian submersion , where , the connected component of the identity in the Lie group , is isomorphic to the full group of orientation-preserving isometries of . The study of the volume of a hyperbolic orbifold is thereby reduced to the study of the covolume of a lattice in a Lie group.
In , Wang showed that the covolume of a lattice in a semisimple Lie group that contains no compact factor can be bounded below by the volume of ball with a radius that depends only on the group itself. We estimate the sectional curvature of and apply a comparison theorem due to Gunther (see e.g. ), to produce a lower bound for . The following theorem gives our main result.
Theorem 0.1 ()
The volume of a hyperbolic –orbifold is bounded below by , an explicit constant depending only on dimension, given by
Remark 0.2 ()
The next section describes a canonical metric for . Section 2 outlines Wang’s crucial result. In the third section, we derive the curvature formulas for a canonical metric of a semisimple Lie group. These formulas are then used to construct an upper bound for the sectional curvatures of .
For comparison, the fifth section lists several results on hyperbolic volume. Sharp volume bounds for hyperbolic orbifolds are known for dimensions 2 and 3. The hyperbolic 2–orbifold of minimum volume was identified by Siegel  in a theorem closely related to a result on birational transformations of an algebraic curve due to Hurwitz . The analogous result for dimension 3 was proved by Gehring and Martin . A hyperbolic orbifold is: a manifold when does not contain elliptic elements; cusped when does contain parabolic elements; arithmetic when can be derived by a specific number-theoretic construction (see e.g. ). What has been established for higher dimensions relate to these categories and their various intersections.
Intimately linked with hyperbolic volume is the size of symmetry groups of hyperbolic manifolds. Specifically, any bound in one category immediately produces a bound in the other. The quotient of a hyperbolic manifold by its group of orientation-preserving isometries is an orientable hyperbolic orbifold (as long as is not virtually abelian, in which case is infinite). The following corollary is a direct analogue of Hurwitz’s formula for groups acting on surfaces.
Corollary 0.3 ()
Let be an orientable hyperbolic –manifold. Let be a group of orientation-preserving isometries of . Then
The Mostow–Prasad rigidity theorem ,  implies that the group of isometries of a finite volume hyperbolic –manifold can be identified with . Hence, we have the following ‘topological’ version of Corollary 0.3.
Corollary 0.4 ()
Let be a finite volume orientable hyperbolic –manifold. Let be a subgroup of . Then
1 The Canonical Metric of
Let be a Lie group and its Lie algebra. For , the adjoint action of is the -endomorphism defined by the Lie bracket
The Killing form on is a symmetric bilinear form given by
We note here that for all , is skew symmetric with respect to ; i.e.,
A Lie group is called semisimple if the Killing form associated to its Lie algebra is nondegenerate. In this case, there exists a Cartan decomposition such that is negative definite and is positive definite, with bracket laws
A positive definite inner product on is defined by putting
Let denote the identity element of . We identify with , the tangent space of at the identity, and extend to a left invariant Riemannian metric over by left translation. This metric will be referred to as a canonical metric for . When the choice of Cartan decomposition is clear, we denote the associated canonical metric by and the induced distance function on by .
Let denote the maximal compact subgroup of with Lie algebra . Important in what follows is that the restriction of to induces a Riemannian metric on the quotient space, as well. In Definition 1.5, the canonical metric on a specific Lie group is scaled in order to secure desired curvature properties for .
Denote by the group of real nonsingular -by- matrices. The Lorentz group is defined by
The Lorentz group is a matrix Lie group; it is a differentiable manifold where matrix multiplication is compatible with the smooth structure. The positive special Lorentz group is the identity component of . It consists of the elements of that have determinant and a positive coordinate.
The Lie algebra of any matrix Lie group is the set of matrices such that , for all real numbers . Denote by the Lie algebra of . Then
Let be an -by- matrix. If , then has the form
For each , let represent the -by- matrix with in the -position and everywhere else. Let and
Definition 1.1 ()
The standard basis for , denoted by , consists of the following set of matrices:
The Lie bracket of a matrix Lie algebra is determined by matrix operations.
The following proposition describes the Lie bracket of .
Proposition 1.2 ()
The proof of the first equation is given here. The proofs of the remaining identities are similar.
By the definition of and the fact that ,
Proposition 1.2 illustrates a Cartan decomposition , where
We note here that , the Lie algebra of the Lie group . In turn, is a maximal compact subgroup of .
In this article, the canonical metric for refers to the canonical metric induced by the Cartan decomposition of (1.8). It is denoted by . The following lemma and corollary give a description of the metric .
Lemma 1.3 ()
Let . Then
The proof follows from a close study of Proposition 1.2.
The set is closed under the Lie bracket (modulo sign). Therefore, for any the entries of are all or and each column has at most one non-zero entry. Since bracket multiplication is determined by index, each row also has at most one non-zero entry. Furthermore, two standard basis elements have a non-zero Lie bracket if and only if they share exactly one index number. So if has index , has exactly
non zero entries.
Now assume . For all This implies that the entry of is the negative of the entry.
The th diagonal entry of is the dot product of the th row of with the th column of . If the only non-zero entry in the th row of is a (resp. ) in the -position then the only non-zero entry in the th column of is a (resp. ) in the -position. Hence, the th diagonal entry of is . Thus,
Let , with . If has a nonzero entry in the -position then the bracket of with the th basis element is sent to the th basis element. That is, there exists such that
If, in addition, has a nonzero entry in the -position, we may write
Again, note that the Lie bracket of basis elements is determined by index. This forces
and we have a contradiction. Thus, all the diagonal entries of are equal to zero. Therefore . ∎
Corollary 1.4 ()
The matrix representation for , the canonical metric for , is the square diagonal matrix
We will be interested in the metric that induces constant sectional curvature on the quotient space . To this end, we scale the metric by the factor .
Definition 1.5 ()
Let be the canonical metric for . The metric on is defined by
2 Discrete Subgroups of Semisimple Lie Groups
For any Lie group, a theorem of Zassenhaus  guarantees the existence of a neighborhood of the identity such that the subgroup generated by any subset of is either non-discrete or nilpotent. Such a neighborhood is called a Zassenhaus neighborhood.
Kazhdan and Margulis  proved that if is a semisimple Lie group without compact factor it contains a Zassenhaus neighborhood such that, for any discrete subgroup of , there exists with the property that . This implies that the fundamental domain for any lattice in has a definite size.
In , H. C. Wang undertook a quantitative study of a Zassenhaus neighborhood for a semisimple Lie group , with respect to a canonical metric. Wang found a value such that a metric ball in centered at the identity with radius satisfied the conclusion of the Kazhdan-Margulis theorem.
Recall the definitions and notations of Section 1. Again, let be a Lie group, its Lie algebra, a Cartan decomposition and the associated inner product. Define a norm on by . For each -endomorphism , let
The number is defined to be the least positive zero of the real-valued function
The following theorem (Theorem 3.2 in ) demonstrates the role of the value in the construction of a Zassenhaus neighborhood for a semisimple Lie group.
Theorem 2.1 (Wang)
Let be a semisimple Lie group. Let denote the identity. Then for any discrete subgroup of G, the set
generates a nilpotent group.
Now, let be the totality of elements in such that the imaginary parts of all the eigenvalues of lie in the open interval and let . In an earlier work , Wang had proved that the restriction of the exponential map to is injective. Hence, the following proposition (Proposition 5.1 in ) establishes the fact that is less than the injectivity radius of .
Proposition 2.2 (Wang)
Let be a semisimple Lie group. Then the closed ball
is contained in .
We now give Wang’s quantitative version of the theorem of Kazhdan-Margulis (Theorem 5.2 in ). It shows that the volume of the fundamental domain of is larger than the volume of a -ball with radius .
Theorem 2.3 (Wang)
Let be a semisimple Lie group without compact factor and . Then for any discrete subgroup of G, there exists such that .
3 The Sectional Curvatures of
In this section, we construct an upper bound on the sectional curvatures of . As a first step, we derive the curvature formulas for a canonical metric of a semisimple Lie group. These formulas are of independent interest as we could not find them in the literature.
A connection on the tangent bundle of a manifold can be expressed in terms of a left invariant metric by the Koszul formula. For any left invariant vector fields , we have
The curvature tensor of a connection is defined by
When a Lie group is semisimple and compact, the canonical metric is the negative of the Killing form and induces a biinvariant metric on . The connection and curvature can be described in terms of the Lie bracket in a simple way (see e.g. [5, Cor. 3.19]).
When is semisimple and noncompact, a canonical metric is biinvariant only when restricted to , the maximal compact subgroup of with Lie algebra . The connection and curvature formulas for this case are given below.
We will treat vector fields from and separately. From here on, and denote left invariant vector fields.
Lemma 3.1 ()
With respect to the canonical metric the subgroup is totally geodesic in .
Now we compute the connections.
Lemma 3.2 ()
The first equation follows from Lemma 3.1. We will derive the last two equations, the proof of the second equation is similar.
The following proposition gives the corresponding curvature formulas.
Proposition 3.3 ()
We prove (3.11). The proofs of the remaining equations are similar.
For a Lie group , with Lie algebra and , the sectional curvature of the planes spanned by and is denoted and defined by
In the next two propositions, we develop our bound for the sectional curvatures of . Recall the notation established in Section 1.
Proposition 3.4 ()
The sectional curvature of with respect to the metric at the planes spanned by standard basis elements is bounded above by .
Proposition 3.5 ()
The sectional curvatures of with respect to are bounded above by
Remark 3.6 ()
4 Volumes of Hyperbolic –Orbifolds
Let and be Riemannian manifolds and a surjective submersion. For each point the tangent space decomposes into the orthogonal direct sum
The map is said to be a Riemannian submersion if
Lemma 4.1 ()
Let denote a fiber bundle where is a Riemannian submersion and is a compact and totally geodesic submanifold of . Then for any subset ,
Since is totally geodesic, the fibers of are isometric to each other. Therefore,
Hence, . ∎
be the quotient map. Recalling the definitions and notations of Section 1, equip with the scaled canonical metric . Furthermore, assign to the quotient the metric induced by the restriction of to . The map is then a Riemannian submersion.
O’Neill’s formula (see e.g. [10, Page 127]) relates the sectional curvature of the base space of a Riemannian submersion, , with that of the total space, . Let represent orthonormal vector fields on as well as their horizontal lifts. O’Neill’s formula, applied to , gives
Here, denotes the vertical component of .
Since and , we have
It follows that the quotient space , with respect to the restriction of the scaled canonical metric, has constant sectional curvature
It can therefore be identified with hyperbolic space, .
For a discrete group , let be the hyperbolic –orbifold defined by the quotient . The map induces another Riemannian submersion
The fibers of on the smooth points of are totally geodesic embedded copies of . By Lemma 4.1, we have