On the volume of orbifold quotients of symmetric spaces
Introduction
In [14], Kazhdan and Margulis proved that every semisimple Lie group without compact factors contains a Zassenhaus neighbourhood. That is, the Lie group G contains a neighbourhood U of the identity, such that any discrete subgroup has a conjugate that intersects U trivially. H. C. Wang studied the size of U with respect to a metric induced by the Killing form on the Lie algebra of G. In [19], Wang associates to each group a positive real number , the value of which depends on two constants and (see Definition 3.1), that in turn are derived from the root system of, and a choice of inner product on, the Lie algebra of G. Wang proves that the volume of the fundamental domain of any discrete subgroup Γ is bounded below by the volume of a ball of radius . For a precise statement of H. C. Wang's result, see Theorem 6.1.
An appendix in [19] lists values for and for the classical non-compact simple Lie groups. However, there is no explanation of how the entries of the table were derived, and hence it was unclear from the paper how one might deduce the values of the constants for the exceptional Lie groups. In this article, we redefine the constants and in terms of a renormalized Killing form (see Section 4). This is the Killing form divided by twice the dual Coxeter number of the Lie group for each simple factor. Our choice reveals a consistency for the values of the that is obscured when the Killing form is used. The values of and for all non-compact simple Lie groups are then deduced using roots and weights theory, and the maximal abelian subalgebra theorem for s-representations.
Theorem 4.5 Let be a simply connected irreducible symmetric space of non-compact type other than hyperbolic 3-space. Equip G 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 1 symmetric space other than , , or , for ; , ; ; or, .
When G is a classical Lie group and is the corresponding symmetric space of type III, the results of Theorem 4.5 agree with those in the appendix of [19], after adjusting for the Killing form. The results for the exceptional Lie groups are new.
Once determined, the values of and also allow for a uniform, dimension independent, upper bound for the sectional curvatures of the metric on a given Lie group. The sectional curvatures are bounded above by a polynomial, with coefficients determined by and , which can then be maximized. We believe this bound to be the sharpest result known.
Theorem 5.3 Let be a real semisimple Lie group with Cartan involution θ that corresponds to an irreducible symmetric space of noncompact type. Let and be the constants given by Definition 3.1, with respect to the inner product defined in Remark 3.3. Let . If the sectional curvatures of G are bounded above by . If the sectional curvatures of G are bounded above by .
Theorem 4.5 and Theorem 5.3 combine to produce a unified treatment of orbifold volume bounds for non-compact irreducible symmetric spaces.
If the quotient of a semisimple Lie group by a discrete subgroup has a finite measure that is invariant under the action of G, then Γ is called a lattice. Fix a left-invariant Haar measure on G. Then the volume of can be given a value for all lattices Γ in a consistent manner. For symmetric space , and lattice Γ of G, the double coset space is an orbifold. It is a manifold when Γ is torsion free. If we choose a left G-invariant, and right K-invariant, inner product on G, then we get a fixed left-invariant Haar measure on G 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. [15]), the volume of orbifold quotients of symmetric spaces are then determined by . As was mentioned above this factor is bounded below by the volume of a ball of radius in G. Typically, the group G will have variable curvature, including both positive and negative values. Our upper bound on sectional curvature, along with a comparison theorem due to Günther [12], produces a lower bound for the volume of the ball.
Theorem 6.3 Let be a non-compact irreducible symmetric space of dimension N, let be the metric on induced by the multiple of the Killing form such that the Ricci curvature is , and let Γ be a lattice of G. Then where, for the Lie group G, is twice the dual Coxeter number, k is an upper bound for the sectional curvature, d is the dimension, and r is the radius of a ball that can be embedded in the fundamental domain of Γ. Furthermore, denotes the metric on induced by times the Killing form, and denotes the gamma function.
The (real) hyperbolic 2-orbifold of minimum volume was identified by Siegel [17] in 1945. The corresponding result for 3-orbifolds was proved by Gehring and Martin [10] in 2009. Recently, Emery and Kim determined the quaternionic hyperbolic lattices of minimal covolume for each dimension [9]. François Thilmany proved the corresponding result for [18]. 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 obtained by several authors. A computable lower bound for the covolume of lattices of a given semisimple Lie group without compact factor, based on the Margulis constant (which itself is difficult to estimate), was proved by Gelander in [11]. For more results in this area, we refer the reader to Section 5 in [1] and Section 4 in [2].
The techniques and results of this paper generalize, and improve upon, those of [1] and [2], in which the sectional curvatures of and were bounded, and hyperbolic and complex hyperbolic volume bounds were derived. For other symmetric spaces, our bound compares favourably to existing results in cases where a minimum value is not known. In particular, we have compared our bound with that of [11] for hyperbolic 4-orbifolds, using the estimate of the Margulis constant in [3], and found that our bound is an improvement (see Section 6).
Section 2 describes the common geometry of symmetric spaces and sets notation for what follows. In Section 3, Wang's definition of the constants and is given, followed by our refinement. We make an observation that simplifies calculation and explain our choice of renormalization of the Killing form. The core of this article is Section 4, where the constants and are determined for each simply connected irreducible symmetric space of non-compact type. Section 5 contains the proof of our upper bound for the sectional curvatures of semisimple Lie groups. Finally, Section 6 provides our application to orbifold volume bounds.
Section snippets
Symmetric spaces
In this section we assemble basic facts on Lie theory that will be used throughout the rest of the paper. Further details on the concepts discussed here can be found in Besse [5] and Helgason [13].
Let denote the Lie algebra associated to a Lie group G. For , the adjoint action of X is the -endomorphism defined by the Lie bracket, that is; The Killing form on is the symmetric bilinear form given by
By Cartan's criterion, a Lie algebra , and
The constants and
The main ingredient for both an upper bound for the sectional curvature of a semisimple Lie group G, and an explicit positive lower bound for the volume of an orbifold , is the evaluation of the values of and for G. These values depend on the root system of, and a choice of metric on, the Lie algebra of G.
Let be a semisimple Lie algebra. The length of is given by , where the inner product is that defined in Section 2. The norm of an endomorphism is defined by
Determination of the constants and
In this section we derive the constants and for all irreducible simply-connected symmetric spaces of non-compact type, or equivalently, for all simple non-compact real Lie groups. Our application for Lie group sectional curvature bounds are given in Section 5 and those for orbifold volume bounds in Section 6. We employ the inner product given in Remark 3.3. In doing so we find that the constants can be computed in a uniform manner which also explains why their ratio takes on only two
The sectional curvatures of semisimple Lie groups
Given a semisimple 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 [1].
Proposition 5.1 In particular,
Remark 5.2 These formulas also apply to any
Orbifold volume bounds
In this section we put together the results of the previous sections and derive explicit orbifold volume bounds. Our calculations make this essentially straight-forward for any orbifold quotient of a symmetric spaces of non-compact type. We will exhibit the bounds in some cases of interest and make some comparisons between our bounds and known bounds in the literature. To do so, we fix a normalization of the metrics used to compute volumes. The standard choice in Riemannian geometry in
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MW was partially supported by NSERC grant OPG0009421.