On the volume of orbifold quotients of symmetric spaces
A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang’s subsequent quantitative analysis showed that the fundamental domain of any lattice contains a ball whose radius depends only on the group itself. A direct consequence is a positive minimum volume for orbifolds modeled on the corresponding symmetric space. However, sharp bounds are known only for hyperbolic orbifolds of dimensions two and three, and recently for quaternionic hyperbolic orbifolds of all dimensions.
As in  and , this article combines H. C. Wang’s radius estimate with an improved upper sectional curvature bound for a canonical left-invariant metric on a real semisimple Lie group and uses Gunther’s volume comparison theorem to deduce an explicit uniform lower volume bound for arbitrary orbifold quotients of a given irreducible symmetric spaces of non-compact type. The numerical bound for the octonionic hyperbolic plane is the first such bound to be given. For (real) hyperbolic orbifolds of dimension greater than three, the bounds are an improvement over what was previously known.
Key words and phrases:Symmetric spaces, lattices, orbifold, volume
2010 Mathematics Subject Classification:Primary 53C35, 22E40
A Riemannian manifold is a symmetric space if for each there exists an isometry of that fixes , and acts on the tangent space of at by minus the identity. Let be the identity component of the group of isometries of . The isotropy subgroup at a point consists of the elements of that fix . The manifold is then diffeomorphic to the quotient . This article considers simply-connected symmetric spaces of non-compact type that are irreducible; that is, not a product of symmetric spaces. In this context, is a non-compact, simple real Lie group and is a maximal compact subgroup.
Let be a discrete subgroup. If the quotient has a finite measure that is invariant under the action of , then is called a lattice. Fix a left-invariant Haar measure on . Then the volume of can be given a value for all lattices in a consistent manner. A result, due to Kazhdan and Margulis , states that for every semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices.
For symmetric space , and lattice of , the double coset space is an orbifold. It is a manifold when is torsion free. If we choose a left -invariant, and right -invariant, inner product on , then we get a fixed left-invariant Haar measure on as well as a Riemannian submersion that is an orthogonal projection on the tangent spaces. Hence, . With the volume of compact Lie groups well understood (see e.g. ), the study of the volume of orbifold quotients of symmetric spaces is equivalent to the study of covolumes of lattices.
First examples of symmetric spaces of non-compact type are the hyperbolic spaces for each dimension: . Complex hyperbolic space , quaternionic hyperbolic space , and the octonionic hyperbolic plane are the analogs of hyperbolic space defined over each of the remaining normed division algebras. Together, the hyperbolic spaces exhaust the category of non-compact symmetric spaces of rank one.
The (real) hyperbolic 2-orbifold of minimum volume was identified by Siegel  in 1945. The corresponding result for 3-orbifolds was proved by Gehring and Martin  in 2009. Recently, Emery and Kim determined the quaternionic hyperbolic lattices of minimal covolume for each dimension . François Thilmany proved the corresponding result for . These remain the only cases where the minimum orbifold volumes are known. Under various geometric and algebraic constraints (e.g. manifold, cusped, arithmetic) lower bounds for the volume of orbifold quotients of the hyperbolic spaces have been constructed by several authors. Section 7 provides an overview of these results.
This article presents a unified treatment of orbifold volume bounds for non-compact irreducible symmetric spaces. In particular, the octonion hyperbolic case is addressed, as well as spaces of higher rank. For a given semisimple Lie group without compact factors, a computable lower bound on the covolume of lattices, based on the Margulis constant, was produced by Gelander in . The techniques of this paper are those of  and , in which hyperbolic and complex hyperbolic volume bounds were derived by combining volume comparison methods in Riemannian geometry with H. C. Wang’s quantitative version of the Kazhdan-Margulis theorem. Our results here in particular improve the bounds of  and , and do not require an estimate of the Margulis constant. We have also compared our bound with that of  for hyperbolic -orbifolds, using the estimate of the Margulis constant in , and found that our bound is an improvement.
In , Kazhdan and Margulis proved that every semisimple Lie group without compact factors contains a Zassenhaus neighborhood of the identity; that is any discrete subgroup in has a conjugate that intersects trivially. H.C. Wang undertook a quantitative study of Zassenhaus neighborhoods in . Wang associates to each group a positive real number . The value of depends on two constants and , which in turn are derived from the root system of, and a choice of inner product on, the Lie algebra of . Using the canonical metric mentioned above, Wang proves that the volume of the fundamental domain of any discrete group is bounded below by the volume of a ball of radius . An appendix in  lists the values of and for the classical non-compact simple Lie groups.
Equipped with the radius , a lower bound for the volume of the ball in Wang’s theorem can be computed from an upper bound on the sectional curvature of the canonical metric using a comparison theorem due to Gunther . In this article, the values of and for the exceptional non-compact simple Lie groups are computed (see Theorem 4.6) together with those of the classical groups in a uniform manner. In addition to their role in determining , the values of and allow for a uniform estimate of the sectional curvatures of our canonical metric on . As a result, volume bounds for all orbifold quotients of symmetric spaces of non-compact type can be calculated.
2. Symmetric Spaces
Let be a Lie group and let denote the associated Lie algebra. For , adjoint action of is the -endomorphism defined by the Lie bracket;
The Killing form on is the symmetric bilinear form given by
which is -invariant.
By Cartan’s criterion, a Lie algebra , and corresponding Lie group , is semisimple if and only if the Killing form on is nondegenerate. A Cartan decomposition for a semisimple Lie algebra is a decomposition
where , , and . Equivalently, one may specify an involutive automorphism of , in which case and are respectively the and eigenspaces of . A semisimple Lie algebra may admit more than one Cartan decomposition. In what follows, we are considering semisimple Lie groups together with a fixed Cartan decomposition. In this case, the Killing form induces a positive definite inner product (left-invariant Riemannian metric) on given by
Then we have . This inner product is -invariant and makes the projection into a Riemannian submersion with totally geodesic fibres. We note here that in Sections 3 and 4, we will abuse notation and use to denote the analogous metric on defined by a more convenient scalar multiple of .
The canonical identification of the Lie algebra of a Lie group with its tangent space at the identity, extends the inner product to a Riemannian metric on by left translation. The induced distance function, referred to as canonical distance, is denoted by .
The objects of our study are irreducible, simply connected symmetric spaces of non-compact type. These spaces are always quotients , where is a real simple non-compact Lie group, and is a maximal compact subgroup of . The restriction of to induces a -invariant Riemannian metric on . Such a space is of type III if , the complexification of , is simple as a complex Lie algebra, and of type IV if not. Note that hyperbolic 3-space has both a type III, , and a type IV, , representation.
In what follows, we will make significant use of the correspondence between a symmetric space of non-compact type and its compact dual . The construction of the Lie group from is as follows: Let be the simply connected complex Lie group that corresponds to . If , then is a (real) Lie subalgebra of . Let be the subgroup of generated by . Since the Killing form is negative definite on and , is compact and contains . A compact symmetric space is of type I or of type II if it is dual to, respectively, a space of type III or type IV.
Type II symmetric spaces have the form . That is, for some compact simple Lie group , and the quotient space is formed with the diagonally embedded subgroup , also identified with . Tables of symmetric spaces, classified by type, can be found in [8, p. 201–202] and [25, p. 518].
The adjoint representation of a Lie group , , sends an element of to the derivative at the identity of the corresponding inner automorphism of . Note that is an isometry of (any multiple) of the Killing form, as well as a Lie algebra automorphism.
The isotropy representation of a homogeneous space at a point is the infinitesimal linear action of on the tangent space at . Isotropy representations of symmetric spaces are often called s-representations. For or , s-representations can also be described as restrictions of the adjoint action of (resp. ) to on (resp. ).
3. The Constants and
The main ingredients in our determination of a lower bound for the volume of are, an explicit positive lower bound for the size of a fundamental domain of , and an upper bound for the sectional curvature of . Both quantities are determined by two values, and , that in turn depend on the root system of, and a choice of metric on, the Lie algebra of .
The length of is given by , where the inner product is that defined in Section 2. The norm of an endomorphism is defined by
In , H. C. Wang defined the constants and as follows:
We give here our refinement of the definitions for and . As indicated in the previous section, the group acts on and , respectively, by the isotropy or adjoint representation. Recall that for the adjoint representation of the compact Lie group , each element of is conjugate to an element in a fixed maximal abelian subgalgebra of . Let and choose such that lies in the preselected maximal abelian subalgebra . Let be any element of . Then,
Note that as runs through all values of , so does .
Less well known is that s-representations, which are examples of polar actions (see ), also admit real maximal abelian subalgebras that contain an element from each orbit. Therefore, a similar calculation holds for . Hence we have,
In determining the constants and , it is sufficient to restrict our attention to the norm one elements lying in a fixed but arbitrary maximal abelian subalgebra of for and of for .
Theorem 3.2 (Wang ).
Let be a semisimple Lie group without compact factors, let be the identity of , let be the canonical distance function, let be the least positive zero of the function
Then for any discrete subgroup of , there exists , such that .
It follows that the volume of , for any , is larger than the volume of the -ball in with radius . Let denote the volume of a ball of radius in the complete simply connected Riemannian manifold of dimension with constant curvature . The following comparison theorem is Theorem 3.101 in .
Theorem 3.3 (Gunther, see ).
Let be a complete Riemannian manifold of dimension . For , let be a ball that does not meet the cut-locus of . Let be an upper bound for the sectional curvatures of . Then,
For fixed, positive values of and , we have by explicit calculation
Therefore, a lower bound for the volume of can be computed by using the dimension of for , the value for , and an upper bound for the sectional curvatures of for . A bound for sectional curvatures, in terms of and , is established in Section 5.
4. Determination of the Constants and
In this section we explain how to determine the constants and that are defined and used for curvature estimates in Section 5, and then for volume bounds in Section 6. The appendix to  includes a table of and for the non-compact classical Lie groups. It is therefore necessary to determine these constants for the exceptional Lie groups as well. In doing so we find that all these constants can be computed in a uniform manner which also explains why their ratio takes on only two possible values.
By definition, the values of the depend on the Cartan decomposition of the Lie algebra , and on the inner product defined in §2. Recall that roots are vectors in the dual of the Cartan subalgebra, and we compute their lengths using the bilinear form on the dual of induced by the Killing form. In this section and the next, we use a renormalized Killing form to define where is the multiple of the Killing form such that the maximal root of has length . This choice reveals a consistency for the values of the that is obscured when the Killing form is used.
4.1. Type IV
We discuss first the relatively straight-forward case of type IV irreducible symmetric spaces.
Let be a compact, simply connected, simple Lie group, and be the corresponding non-compact symmetric space. Equip with the left-invariant metric described above. Then .
In this case the compact form is . The corresponding Cartan decomposition of is the sum of the diagonal of , with the anti-diagonal in . Hence the Lie algebra of the non-compact dual is
Hence, the adjoint and isotropy actions of are equivalent and we have that .
To compute , we already saw that we only need to consider the norms of operators , where ranges over norm elements in a fixed maximal abelian subalgebra in . But such an algebra is exactly times a maximal abelian subalgebra in , which is the tangent space of a maximal torus in . In other words, is just a real Cartan subalgebra in the complex simple Lie algebra . For , the action of on can be read off from a Weyl-Chevalley basis of .
In fact, starting from the compact group , with inner product and a choice of maximal torus and weak order, one can construct a Weyl-Chevalley basis of the form
where is the set of positive roots of and is its rank. This basis has the following properties:
The are orthonormal with respect to ;
are orthonormal with respect to ;
, where is the element in dual to with respect to .
Let , where and . Then,
If we expand in terms of the Weyl-Chevalley basis and apply (a)-(d), and use the fact that
we obtain . This shows that . However, the upper bound is realized by setting and where is the maximal root of . This completes the proof of the proposition. ∎
We now compare the values of obtained above with those in the first three rows of the appendix in 
First, note that the Killing form of induces on the Killing form of . On the other hand, for the metric we use on , it is the negative of the Killing form on . This accounts for the difference of a factor of between our values of and those in the first three rows of the table in .
Furthermore, to obtain the values of with respect to the Killing form rather than the renormalized Killing form, one needs to divide by the constant defined by
A table for the normalizing constants can be found, for example, in [40, p. 583].
With the above adjustments, the values we obtained agree with those (in the first three rows of the appendix) in , except for a possible typographical error there in the value of for the groups of BD type: the factor of 4 in the denominator should be a instead.
We now discuss the case of irreducible symmetric spaces of type III in two parts.
4.2. in Type III
Let be an irreducible symmetric space of type III. If the constant , with respect to the inner product defined using the renormalized Killing form, is equal to . For all other cases .
Assume . We fix a maximal abelian subalgebra and a weak ordering in . As before, it is enough to consider the norms of for norm elements . Now acts on as the direct sum of the adjoint and isotropy representations, the latter of which is irreducible over . If an -irreducible summand in these representation split upon complexification, the resulting complex irreducible summands are uniquely characterized by a pair of maximal roots (resp. dominant weights ) which have the same norm. Since a dual pair of irreducible symmetric spaces have isomorphic isotropy representations, it follows that we can work with the type I compact dual spaces as far as is concerned.
For the adjoint representation of we can choose a Weyl-Chevalley basis as in Proposition 4.1. One sees that the nonzero eigenvalues of occur among the values . For the -part we can take a -orthonormal set in which ranges over all the positive weights of the isotropy representation. We can then define the complex weight vectors which satisfy . Thus the nonzero eigenvalues of occur among the values . It follows from this that if and such that , then As in the proof of Proposition 4.1, both upper bounds are easily seen to be attained when for (resp. ).
Note that the norms and are computed using the renormalized Killing form of and not that of . When and have the same rank, then the roots of and the weights of the isotropy representation are all roots of . Hence one of or equals and we are done. It therefore remains for us to check if these norms equal for with .
Recall that the index of a compact simple Lie subalgebra of a compact simple Lie algebra is the positive integer so that . The index allows us to compare norms defined using the renormalized Killing forms of and . We refer the reader to  for details and p. 584 of  for a summary of the pertinent facts that we will use.
Among the type I symmetric spaces with unequal rank and simple ; , , and , the index . To analyze these cases we will use the standard parametrizations of the root systems of , and knowledge of the isotropy representation (see for example pp. 324-325 of ). In the first case, and , so . In the second case, while , and so In the third case, which is shorter than the maximal root of .
Two cases remain: and with or . (Note that the subcase of the former is the same as the subcase of the latter.) In the first case, the maximal root of is , which is shorter than the dominant weight of the isotropy representation. However, , so has norm with respect to the renormalized Killing form of . One needs to treat the special cases ( is non-simple) and () separately but the conclusion is the same.
There are a number of special cases within the second case. If , both simple factors in have index . There are two maximal roots and both of norm with respect to the renormalized Killing form. The dominant weight of the isotropy representation is which also has norm . So . If , is , the dual of hyperbolic space, which is treated in . With the current point of view, assuming , we have . The dominant weight of the isotropy representation is , which is shorter than the maximal root of of length . Again we have . For the remaining cases, either or equals . The differences are that for and the nonzero roots of are . With these changes one still gets .
Finally, the case of hyperbolic -space, with corresponding compact dual is special because and the maximal root of has length only with respect to the renormalized Killing form of . Thus in this special case, as noted in . It is also interesting to note that can be written as the type IV symmetric space . The difference in the values of is due to the difference in the renormalized Killing forms. ∎
The case of hyperbolic -space is also special. The compact form is , so is abelian and has no nonzero roots. However, , so with respect to the renormalized Killing form of . Note that in  the metric used in defining is twice the renormalized Killing form of , so the corresponding value of becomes .
For complex hyperbolic space, the compact dual is projective space . In , the metric used to define is twice the renormalized Killing form of , so instead.
4.3. in Type III
So far we have only made use of the structural and representation theory of compact or complex semisimple Lie groups. We now appeal to elements of the theory of restricted roots for symmetric spaces. First, we reintroduce some notation and recall basic facts from pp. 257 - 263 of .
Given a type III symmetric space with involution and Cartan decomposition , we fix a maximal abelian subalgebra of lying in . As remarked before, we consider the norms of for elements of norm with respect to the metric defined using the renormalized Killing form of .
Denote by a fixed extension of to a maximal abelian subalgebra of . The -invariant algebra can be decomposed as where is a maximal abelian subalgebra in . The complexification of is a Cartan subalgebra of the complex Lie algebra and is the corresponding real Cartan subalgebra. The roots of are regarded as elements of the dual of . We will also fix a compatible weak ordering of elements in and to get a consistent notion of positivity for roots and restricted roots.
Let be the dual compact form of . Then and the maximal abelian subalgebra is the Lie algebra of a maximal torus in . As in the proof of Proposition 4.1, we choose a Weyl-Chevalley basis of having the stated properties there.
Let denote the positive roots of . Upon restriction to , some of the roots will vanish. These are often called the compact roots and they are characterized by being fixed by the action of the Cartan involution . Those roots which do not vanish identically may be called noncompact roots and their restrictions to , viewed as elements of , are called restricted roots. We let (resp. ) denote the set of positive noncompact (resp. restricted) roots. Their relevance to the determination of can now be explained.
Choose of norm . By Lemma VI.1.2 in , is self-adjoint with respect to the metric , which is denoted by in . The norm of as an operator on is the absolute value of its largest eigenvalue. We can analyse its eigenvalues by looking at its complex extension to , where we can use our Weyl-Chevalley basis. It follows that
The constant is therefore equal to the supremum of these norms over the norm one elements of . An upper bound for is clearly since we are using the renormalized Killing form. But except in the case of hyperbolic -space. Thus and in the case of hyperbolic -space we have , for example by .
We can now go through the list of type III symmetric spaces, examine their restricted root systems, and deduce the values of . A convenient source for this purpose is Table VI, pp. 532 - 534 of . We will exclude the special case of hyperbolic -space in the following discussion.
Suppose and we know that the dual element . If in addition is a long root of (this is automatic if is of ADE type), then since . If all elements of are short roots, then .
Let be a simply connected irreducible symmetric space of non-compact type other than hyperbolic -space. Equip with the left-invariant metric induced by the renormalized Killing form. Let be the constants defined in Section 3. Then either or . The latter occurs exactly when is one of the following:
A rank symmetric space other than , , or , for ;
The first case to which Observation 4.4 applies is that of normal real forms. This means that the maximal abelian subalgebra is actually maximal abelian in , and so can be taken as a real Cartan subalgebra of . In particular, so it always contains the maximal root. Hence . The entries in Table VI of  belonging to this case are: AI, ; BI, ; CI, ; DI, ; EI; EV; EVII; FI; and G. Note that , corresponding to hyperbolic -space or complex hyperbolic -space, also belongs here as has no non-zero roots.
Observation 4.4 is also applicable when is of ADE type and there is a restricted root whose multiplicity is odd, by F3 on p. 530 of . Besides the normal real forms we have AIII, ; DI, ; DIII, ; EII; EIII; EVI; EVII; and EIX. In all these cases, again . The same conclusion also holds for the case BI: , odd, , since there are restricted roots which come from long roots whose dual coroot lies in .
For the case of the Cayley plane , take as root system for
A system of positive simple roots is (see , p. 80, for this choice). In the Satake diagram, a blackened vertex means the corresponding root restricts to zero for the chosen maximal abelian subalgebra . So is given by , i.e., is spanned by the short root . The restricted roots are with respective multiplicities of and . The restriction of the renormalized Killing form on our real Cartan subalgebra is just the Euclidean inner product. With and restricted root , we have or . Hence by Equation (4.1), .
For the split rank case of EIV, , we use the root system for given on p. 80 of . The positive roots are
with fundamental system
The renormalized Killing form is From the Satake diagram we see that is defined by
so that . For elements of the renormalized Killing form becomes . The restricted roots are , all with multiplicity . One then deduces that the maximum value of these linear forms subject to the norm condition is in all cases. Thus .
The remaining symmetric spaces of type III that need to be examined are AII: , and CII: . (Recall that the real hyperbolic space case was treated in .)
For the case of ), , we use the usual parametrization of the real Cartan subalgebra of . The Satake diagram gives to be
The positive restricted roots can be taken to be , each with multiplicity . The dual renormalized Killing form is such that the linear forms are orthonormal. In particular, an element of has norm if and only if . One easily checks that the maximum value of on norm elements of is . For this is realised by the element . Hence . Note that the norm element in the Cartan subalgebra produces the value with (which gives the restricted root ), but it does not lie in .
Similar computations for the CII case, which we leave to the reader, show that as well. ∎
When is a classical Lie group and is of type III the above values of agree with those in the appendix of  after adjusting for the Killing form. For convenient reference, we summarize the results for the exceptional Lie groups developed in this article in the following proposition.
Let be a simply connected irreducible symmetric space of non-compact type with an exceptional Lie group. Then with respect to the metric induced by the renormalized Killing form, the constants , except when is the Cayley projective plane or , in which case .
5. The Sectional Curvatures of Semi-Simple Lie Groups
Given a semi-simple Lie algebra with Cartan decomposition , let and denote left invariant vector fields. The curvature formulas for the canonical metric of a semisimple non-compact Lie group were derived in .
These formulas also apply to any scale of the canonical metric.
If the sectional curvatures of are bounded above by . If the sectional curvatures of are bounded above by .