Orbifolds with Lower Ricci Curvature Bounds

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 125, Number 10, October 1997, Pages 3011–3018 S 0002-9939(97)04046-X

ORBIFOLDS WITH LOWER RICCI CURVATURE BOUNDS

JOSEPH E. BORZELLINO

(Communicated by Christopher Croke)

Abstract. We show that the ﬁrst betti number b1(O)=dimH1(O, R)of a compact Riemannian orbifold O with Ricci curvature Ric(O) (n 1)k ≥− − and diameter diam(O) D is bounded above by a constant c(n, kD2) 0, depending only on dimension,≤ curvature and diameter. In the case when≥ the orbifold has nonnegative Ricci curvature, we show that the b1(O) is bounded above by the dimension dim O, and that if, in addition, b1(O)=dimO,then Ois a ﬂat torus T n.

Introduction In 1946, S. Bochner [Bo] proved that if M is a compact n–dimensional Rie- mannian manifold with nonnegative Ricci curvature, then the first betti number ˜ b1(M)=dimH1(M,R) dim M. Moreover, the universal cover M of M is the ≤ product N Rb1(M). The question naturally arises as to which manifolds admit maximal first betti× number. The answer is that M must be an n–dimensional flat torus T n = Rn/Zn. The original proof is based on the study of harmonic 1–forms and uses the Weitzenböck formula ∆α = D∗Dα+Ric α which expresses the Hodge Laplacian of a 1–form α in terms of its connection Laplacian and Ricci curvature. See [Be]. Later, Gromov [GLP] and Gallot [G] showed that there is a constant c(n, kD2) 0 such that the class of n–dimensional compact Riemannian manifolds with Ricci≥ curvature Ric(M) (n 1)k and diameter diam(M) D have first betti number 2 ≥− − 2 ≤ b1(M) c(n, kD ). Moreover, limkD2 0+ c(n, kD )=c(n, 0) = dim(M). In this≤ paper we generalize these results→ to the class of Riemannian orbifolds. Specifically, we prove the following Theorem 1. Let O be a compact n–dimensional Riemannian orbifold whose Ricci curvature Ric(O) (n 1)k and diameter diam(O) D. Then there is a constant c(n, kD2) 0 depending≥− − only on dimension, curvature≤ and diameter such that ≥ 2 (i) b1(O) c(n, kD ). (ii) c(n, kD≤2)=0for k<0. 2 (iii) limkD2 0+ c(n, kD )=c(n, 0) = n. → orb Remark 2. For orbifolds there is a notion of the orbifold fundamental group π1 (O), and thus one can define a first orbifold betti number as the rank of the abelianiza- orb tion of π1 (O). It turns out, however, that this orbifold betti number agrees with

Received by the editors May 15, 1996. 1991 Mathematics Subject Classiﬁcation. Primary 53C20.

c 1997 American Mathematical Society

3011

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3012 JOSEPH E. BORZELLINO

the standard one (Proposition 9), and hence there is no loss in generality in stating our theorems in terms of the standard betti number b1(O). These matters will be discussed more thoroughly in the next section. We have the following immediate corollary of Theorem 1: Corollary 3. Let O be a compact n–dimensional orbifold whose Ricci curvature Ric(O) (n 1)k and diameter diam(O) D. There exists an ε = ε(n) > 0, such that≥− if kD−2 <ε,thenb(O) n. ≤ 1 ≤ As a consequence of Theorem 1 and Proposition 9 we will be able to generalize Bochner’s Theorem to Riemannian orbifolds. Note that Theorem 4 (i) and (ii) are immediate corollaries of Theorem 1. Theorem 4. Let O be a compact n–dimensional Riemannian orbifold.

(i) If the Ricci curvature is strictly positive, then b1(O)=0. (ii) If the Ricci curvature is nonnegative, then b (O) n. 1 ≤ (iii) If the Ricci curvature is nonnegative and if b1(O)=n,thenOis a good orbifold. In fact, O is isometric to an n–dimensional flat torus. Remark 5. One might expect that in the orbifold situation the equality case would only yield a flat orbifold of the form T n/G where G is some finite group acting isometrically (with possible fixed points) on T n. The fact is, however, that the maximality of the first betti number rules out the possibility of singularities. The author would like to thank the referee for suggesting improvements to the original paper, and Professor Peter Petersen for some helpful discussions.

Orbifold preliminaries The basic reference for orbifolds is [T]. The book [R, Chapter 13] is also a good reference for the notions that appear in this paper. The essential facts we need are orb that π1 (O) is the group of (orbifold) homotopy classes of loops in O,andthat associated to every orbifold O there is a connected universal covering orbifold O˜ orb ˜ orb with π1 (O) = 0 such that the elements of π1 (O) are the deck transformations of ˜ orb O. The group π1 (O) acts (properly) discontinuously and isometrically (but not ˜ ˜ orb necessarily freely) on O and as orbifolds O = O/π1 (O). orb Definition 6. Let π1 (O) be the orbifold fundamental group. We define the first orb orb orbifold homology group H1 (O, Z) to be the abelianization of π1 (O)andthefirst orb orbifold betti number to be dim H1 (O, Z) R . ⊗ 2 3 2 Example 7. Consider the standard 2–sphere S R . Define a Zp–action on S ⊂ 2 by rotation around the z–axis by an angle of 2π/p. The quotient space O = S /Zp orb orb is commonly referred to as a Zp–football. In this case, π1 (O)=H1 (O, Z) ∼= Zp, orb but H1(O, Z) = 0. Note, however, that b1 (O)=b1(O)=0. orb Remark 8. Although, as Example 7 illustrates, H1 (O, Z) is in general not equal orb to H1(O, Z), it is always true that b1 (O)=b1(O). We prove this in Proposition 9. Note that by the construction of the orbifold fundamental group (for example, orb see [HD]) there is an epimorphism ϕ : π1 (O) π1(O) from the orbifold fundamental group to the (standard) fundamental group. This gives rise to the following

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ORBIFOLDS WITH LOWER RICCI CURVATURE BOUNDS 3013

commutative diagram:

orb Ab orb R orb orb π1 (O) H 1 (O, Z) ⊗ H 1 (O, Z) R = H1 (O, R) −−−−→ −−−−→ ⊗ ϕ ϕ˜ ϕˆ

 Ab  R  π1(O) H 1 ( O, Z) ⊗ H 1 ( O, Z) R = H1(O, R) y −−−−→ y −−−−→ ⊗ y 1 where Ab : G G/[G, G] is the abelianization of a group G,and˜ϕ=Ab ϕ Ab− . It’s easy to see→ thatϕ ˜ is well–defined and that all arrows are epimorphisms.◦ ◦ We thus conclude the following Proposition 9. Let O be a Riemannian orbifold. Then the first orbifold homology orb group with real coefficients H1 (O, R)=H1(O, R), and hence the first orbifold betti orb number b1 (O)=b1(O). orb Proof. Letα ˆ Kerϕ ˆ.Ifˆα= 0, then any preimageα ˜ H1 (O, Z) has infinite order orb ∈ 6 orb ∈ in H1 (O, Z). Similarly, let α π1 (O) be any preimage ofα ˜. Commutativity of the diagram implies thatϕ ˜(˜α) has∈ finite order. Thenϕ ˜(˜αm) = 0, which implies that m m orb orb m ϕ(α ) [π1(O),π1(O)], and thus α [π1 (O),π1 (O)]. But then [Ab(α)] =0, which implies∈ thatα ˜ has finite order,∈ which is a contradiction. Thus,α ˆ =0andˆϕ is an isomorphism. This completes the proof.

Growth of the orbifold fundamental group

Let G be a finitely generated group and let S = g1,... ,gk be a system of generators. Each element g of G can be represented by{ a word g}p1 gp2 gp` and i1 i2 ··· i` the number p1 + + p` is called the length of the word. The norm g word (relative to S| ) is| defined··· | as| the minimal length of words representing g. Fork k any t Z+, the number of elements of G which can be represented by words whose length∈ is t will be denoted ϕ (t); that is, ϕ (t) is the number of elements with ≤ S S g word t. A group G is said to have polynomial growth of degree n if for some k k ≤ ≤ n system of generators S there is a constant c>0 such that ϕS(t) c t .Itisnot hard to see that this definition is independent of the choice of generators≤ · S.See [Z]. J. Milnor [M] observed that if M is an n–dimensional compact Riemannian manifold with nonnegative Ricci curvature, then the fundamental group π1(M)has polynomial growth of degree n. We should point out, however, that A. Schwarz [S] was the first to discover the≤ relationship between the growth of the fundamental group and the volume growth of balls in the universal cover M˜ . Essentially, the same argument yields Proposition 10. Let O be a compact n–dimensional Riemannian orbifold with orb nonnegative Ricci curvature. Then the orbifold fundamental group π1 (O) has polynomial growth of degree n. ≤ orb Proof. π1 (O) acts isometrically and (properly) discontinuously on the universal orbifold cover O˜.Letp O˜be a nonsingular point of O˜.Thereexistsr>0such orb ∈ that for g π1 (O) the balls B(g(p),r) are pairwise disjoint. Let S = g1,... ,gk ∈ orb { } be a system of generators of π1 (O). Let

L =maxd(p, gi(p)). i

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3014 JOSEPH E. BORZELLINO

orb If g π1 (O) can be represented as a word of length t with respect to the ∈ p1 p` ≤ gi’s, then g = g g , 1 ` t, with pi = 1. Note that i1 ··· i` ≤ ≤ ± d(p, g(p)) = d(p, gp1 gp`(p)) i1 ··· i` d(p, gp1 (p)) + d(gp1 (p),gp1 gp`(p)) ≤ i1 i1 i1 ··· i` p2 p` L + d(p, g g (p)) since the gi’s are isometries ≤ i2 ··· i` L+L+ +L= L t. ≤···≤ ··· · t times Taking all such g’s, we obtain ϕ (t) disjoint balls B(g(p),r) such that B(g(p),r) |S {z } B(p, Lt + r). Thus, ⊂ ϕ (t) Vol B (p, r) Vol B (p, Lt + r) S · ≤ which implies that Vol B (p, Lt + r) (Lt + r)n ϕ (t) . S ≤ Vol B (p, r) ≤ rn The last inequality follows from the relative volume comparison theorem for orb- orb ifolds [B]. Hence we conclude that π1 (O) has polynomial growth of degree n. This completes the proof. ≤

Rigidity of orbifolds of maximal first betti number In this section we prove Theorem 4 (iii). We now assume that O has maximal first betti number b1(O)=n. By Proposition 10, the orbifold fundamental group orb π1 (O) has polynomial growth exactly = n. By the orbifold generalization of the Cheeger–Gromoll splitting theorem [BZ] we know that the universal orbifold cover O˜ splits as an isometric product N Rk, where N is a compact orbifold with nonnegative Ricci curvature. Also, the isometry× ˜ k orb ˜ group splits, namely, Isom(O) = Isom(N) Isom(R ). Now π1 (O) Isom(O) × ⊂ acts properly discontinuously on O˜. As a result, we have the exact sequence pr 1 F π orb(O) 2 C 1 −→ −→ 1 −→ −→ k where F is finite and C is a crystallographic group acting on R ,andwherepr2 orb k denotes the homomorphism pr2 : π1 (O) Isom(R ) with kernel F . See [BZ]. It then follows that C must have polynomial→ growth of degree = n. By applying (standard) relative volume comparison it follows that k = n;thatis,Cmust act on Rn.ThisimpliesthatO˜=Rn and thus O is the (good) flat orbifold O/C˜ ,and orb C=π1 (O). To finish the proof we use the Bieberbach theorems on crystallographic groups. Let C∗ be the normal subgroup of pure translations of C. The Bieberbach theorems [W, Theorem 3.2.9] imply that any crystallographic group C satisfies the exact sequence n orb 1 ( Z = C ∗ ) C = π (O) H 1 −→ ∼ −→ 1 −→ −→ where H ∼= C/C∗ is a finite group. Recall the following argument in the case that C acts freely (that is, C is a Bieberbach group). In this case, O is a compact flat manifold and C = π1(O). π1(O) is torsion free by [W, Theorem 3.1.3]. We claim in fact that π1(O) is abelian. n Hence, π1(O)=Z , and by the Bieberbach theorems [W, Theorem 3.3.1], O must be a flat torus.

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ORBIFOLDS WITH LOWER RICCI CURVATURE BOUNDS 3015

To show that π1 (O) is abelian we proceed as follows. Suppose not. Then by abelianizing π1(O) some nontrivial element, say a, gets sent to 1. Now the exact k n n sequence implies that any element a π1(O) satisﬁes a Z where we regard Z as a subgroup of π (O). Since π (O∈) is torsion free, ak ∈=1,andak gets sent to 1 1 6 1 under abelianization. Thus, dim H1(O, R)

The first betti number estimates We now prove Theorem 1. For the proof of part (ii), we use the fact that strictly positive Ricci curvature implies that O is compact. Since the universal orbifold cover satisfies the same curvature restrictions as O, we conclude that the universal ˜ orb orbifold cover O is compact and thus π1 (O) is finite, and b1(O)=0.Toseethat strictly positive Ricci curvature implies compactness of O, assume for simplicity that Ric(O) (n 1). We show that diam(O) π. Suppose not and choose ≥ − ≤ points p0,q0with d(p0,q0) >π+ε. We can choose nonsingular points p, q such that d(p, p0) <ε/3andd(q, q0) <ε/3, since the singular set is nowhere dense. Then d(p, q) >π+ε/3. Let γ be a minimal unit speed geodesic joining p to q.Thenγ does not intersect the singular set. See [B]. Thus γ(π) is inside the cut locus of p and hence the distance function from p, d(p, )issmoothatγ(π). By the Laplacian comparison theorem for orbifolds [BZ] we have· that ∆d(p, ) (n 1) cot d(p, ). Letting d(p, ) π fromtheleftimpliesthat∆d(p, ) · ≤, which− contradicts· smoothness of· d→(p, )atγ(π). · ≤−∞ The proof of part· (i) follows closely the proof of the standard case given in [Z]. The first observation to make is that it is sufficient to show that there is a

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3016 JOSEPH E. BORZELLINO

ˆ orb ˆ 2 finite (orbifold) cover O O such that π1 (O) has at most c(n, kD ) genera- → orb ˆ tors. To see this, suppose that h1,... ,hk generate π1 (O). Then the index orb orb ˆ { } m orb ˆ [π1 (O):π1 (O)] = m< , so there are g1,... ,gm such that gi π1 (O) orb m orb ∞ˆ { } ∈ and π1 (O)= i=1 giπ1 (O). As before, let Ab : G G/[G, G] be the abelianization of a group G, and consider the composition: → S orb Ab orb orb π (O) H (O, Z) ⊗R H (O, R) 1 −→ 1 −→ 1 and let f =( R) Ab denote this composition. Since h1,... ,hk,g1,... ,gm orb ⊗ ◦ orb{ } generates π1 (O), the set Ab(hi), Ab(gj ) generates H1 (O, Z). Now note that m { orb }ˆ orb m f(gj )=f(g )=f(h)forsomeh π (O). Thus f(hi) generate H (O, R), · j ∈ 1 { } 1 and hence to bound b1(O) it is sufficient to bound the number of generators of orb ˆ π1 (O). Letπ ˜ : O˜ O denote the universal orbifold cover. Choose a nonsingular point p O,andchoose˜→ p O˜withπ ˜(˜p)=p.Fixε>0. Denote g = d(˜p, g(˜p)). Take a∈ maximal set g ,...∈ ,g of πorb(O) such that k k { 1 m} 1 1 g 2D+ε, and g− g ε, i = j. k ik≤ k i jk≥ 6 orb ˆ Let Γ be the subgroup of π1 (O) generated by gi and letπ ˆ : O O be the orb ˆ { } → orbifold covering of O with π1 (O)=Γ.Toshowthatˆπis a finite cover we show that diam(Oˆ) 2D +2ε.Letˆp Oˆbe such thatπ ˆ(ˆp)=p.Thenˆpis nonsingular. ≤ ∈ If diam(Oˆ) > 2D +2ε, then there is a pointq ˆ Oˆ such that d (ˆp, qˆ)=D+ε. ∈ Oˆ Since d (p, πˆ(ˆq)) diam(O)=D, there is a deck transformation α πorb(O) Γ, O ≤ ∈ 1 − such that d (ˆq,π(αp˜)) D,whereπ:O˜ Oîs the covering projection. Then Oˆ ≤ → d (ˆp, π(αp˜)) d (ˆp, qˆ) d (ˆq, π(αp˜)) ε, Oˆ ≥ Oˆ − Oˆ ≥ d (ˆp, π(αp˜)) d (ˆp, qˆ)+d (ˆq, π(αp˜)) 2D + ε. Oˆ ≤ Oˆ Oˆ ≤ orb ˆ Now note that there is a β π1 (O) = Γ such that dO˜ (Γ˜p, αp˜)=dO˜(βp,˜ αp˜), so ∈ 1 1 β− α = d ˜ (˜p, β− αp˜)=d˜(βp,˜ αp˜)=d˜(Γ˜p, αp˜) k k O O O = d (ˆp, π(αp˜)) 2D + ε. Oˆ ≤ Also, for any g πorb(Oˆ)=Γ, ∈ 1 1 1 1 1 g− β− α =d˜(˜p, g− β− αp˜)=d˜(βgp,˜ αp˜) d ˜ (Γ˜p, αp˜) k k O O ≥ O = d (ˆp, π(αp˜)) ε. Oˆ ≥ 1 So by maximality of Γ, β− α should be in Γ. However, α is not in Γ by construction and β is; therefore we have a contradiction, and we conclude that diam(Oˆ) 2D +2ε. ≤ Recall that g1,... ,gm is a set of generators of Γ. We now give a bound on the the number of{ generators }m in terms of dimension, curvature, and diameter. We denote by B(˜p, r)themetricr–ball in O˜ centered atp ˜ and B(r), the metric r–ball in the simply connected space form of curvature k. Since −

1 1 ε g− g =d˜(g− g p,˜p˜)=d˜(g p,˜ g p˜) ≤k i jk O i j O j i

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use ORBIFOLDS WITH LOWER RICCI CURVATURE BOUNDS 3017

1 m 1 the balls B(gip,˜ 2 ε) are pairwise disjoint. Moreover, we have that i=1 B(gip,˜ 2 ε) B(˜p, 2D + 3 ε). Thus, ⊂ 2 S Vol m B(g p,˜ 1 ε) Vol B (˜p, 2D + 3 ε) m = i=1 i 2 2 Vol B (˜p, 1 ε) ≤ Vol B (˜p, 1 ε) S 2 2 Vol B (2D + 3 ε) Vol B (5D) 2 = c(n, kD2) 1 ≤ Vol B ( 2 ε) ≤ Vol B (D) where in the third to last inequality we have used the relative volume comparison for orbifolds [B] and in the second to last we have chosen ε =2D. This completes the proof of part (i) of Theorem 1. We now proceed to the proof of part (iii) of Theorem 1. We are going to modify orb the previous proof by choosing longer loops to generate H1 (O, R). Again let orb orb f : π1 (O) H1 (O, R) be the map defined in the proof of part (i), and let → orb g1,... ,gb1(O) be a basis for H1 (O, R)chosenasabovewithε=2D.Denote { } orb by Γ the subgroup of π1 (O) generated by the gi’s. If Γ contains an element g =1 with g < 2D,thenf(g)=0andf(g) has infinite order. Thus, there exists6 an r>0k suchk that 2D gr 6 4D.Writef(g)=a f(g )+ +a f(g )=0. ≤k k≤ 1 1 ··· b1(O) b1(O) 6 Without loss of generality we may assume that a1 =0.LetΓ1 be the group r 6 r generated by g ,g2,... ,gb1(O) . Then clearly, the set f(g ),... ,f(gb1(O)) is { orb } { } still a basis for H1 (O, R). Also, g Γ . To see this suppose that 6∈ 1 g = c (gr)k1 c (gr)k2 c (gr)k` c 1 · · 2 · ··· ` · · `+1 where the c ’s are words in g ,... ,g .Then i { 2 b1(O)} f(g)=f(c )+f(grk1 )+ +f(c )+f(grk` )+f(c ) 1 ··· ` `+1 = f(c c c )+r(k + +k )f(g). 1 2 ··· `+1 1 ··· ` Since r 2, we can rewrite the expression above as ≥ 1 f(g)= f(c c c )=a f(g )+ +a f(g ) 1 r(k + +k ) 1 2 ··· `+1 2 2 ··· b1(O) b1(O) − 1 ··· ` which contradicts the assumption that a1 =0. orb 6 orb Since π1 (O) contains only finitely many g with g < 2D (π1 (O) acts discontinuously andp ˜ is nonsingular), after finitely many suchk k replacements we produce a orb new set h1,... ,hb1(O) of elements of π1 (O)sothat f(h1),... ,f(hb1(O)) form { orb } { } a basis of H1 (O, R), and 2D hi 4D.IfΓ0 Γ is the subgroup generated ≤k k≤ ⊂ by the h ’s, then for any element h Γ0 1 ,wehave h 2D.Let i ∈ −{ } k k≥ S(t)= h Γ0 h t . ∈ k kword ≤ 1 If h, h0 Γ0, with h = h0,then h− h0 2D, which implies that the balls ∈ 6 k k≥ B(hp,˜ D),h Γ0, are pairwise disjoint. If b (O) b, then it is not hard to see that ∈ 1 ≥ card(S(t)) t b where card(S(t)) denotes the cardinality of S(t). Since ≥ b B(hp,˜ D) B(˜p, 4Dt + D) ⊂ h S(t) ∈[

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 3018 JOSEPH E. BORZELLINO

we have

Vol B (˜p, 4Dt + D) Vol B (hp,˜ D) = card(S(t)) Vol B (˜p, D) ≥   · h S (t) ∈[  b  t Vol B (˜p, D). ≥ b By applying relative volume comparison for orbifolds again, we see that t b Vol B (˜p, 4Dt + D) Vol B (4Dt + D) b ≤ Vol B (˜p, D) ≤ Vol B (D) 4Dt+D n 1 1/2 √ − (4t+1)D√k n 1 0 k− sinh s k ds sinh − sds = = 0 D n 1 D√k n 1 R 1/2 √ − R sinh − sds 0 k− sinh s k ds 0 =(4tR+1) n + 5ntnif kD2 is smallR enough. ···≤ n Thus, if b1(O) (n + 1), then we set b =(n+1)aboveandﬁx t>5 (n+1)n+1.Nowchoosing≥ kD2 suﬃciently small gives a contradiction, for we would· n+1 have t 5ntn which implies that t 5n (n +1)n+1.Thusb(O) n. n+1 ≤ ≤ · 1 ≤ This completes the proof. References

[Be] A. Besse. Einstein Manifolds, Springer–Verlag, New York 1987. MR 88f:53087 [Bo] S. Bochner. Vector Fields and Ricci Curvature, Bull. Am. Math. Soc. 52 (1946), 776–797. MR 8:230a [B] J. Borzellino. Orbifolds of Maximal Diameter, Indiana U. Math. J. 42 (1993), 37–53. MR 94d:53053 [BZ] J. Borzellino and S. Zhu. The Splitting Theorem for Orbifolds, Illinois J. Math. 38 (1994), 679–691. MR 95c:53043 [Br] G. Bredon. Introduction to Compact Transformation Groups, Academic Press, New York 1972. MR 54:1265 [G] S. Gallot. A Sobolev Inequality and Some Geometric Applications, Spectra of Riemannian Manifolds, Kaigai Publications, Tokyo, 1983, 45–55. [GLP] M. Gromov, J. Lafontaine, and P. Pansu. Structures métriques pour les variétés rieman- niennes, Cedic–Nathan, Paris, 1980. MR 85e:53051 [HD] A. Haefliger and Q. Du. Une présentation du Groupe Fondamental d’une Orbifold,in Structure Transverse Des Feuilletages, Astérisque No. 116, (1984), 98–107. MR 86c:57026b [M] J. Milnor. A Note on Curvature and Fundamental Group, J. Diff. Geo. 2 (1968), 1–7. MR 38:636 [R] J. Ratcliffe. Foundations of Hyperbolic Manifolds, Springer–Verlag, New York 1994. MR 95j:57011 [S] A. Schwarz. A Volume Invariant of Coverings, Dokl. Ak. Nauk. USSR 105 (1955), 32–34 [in Russian]. [T] W. Thurston. The Geometry and Topology of 3–Manifolds, Lecture Notes, Princeton Uni- versity Math. Dept., 1978. [W] J. Wolf. Spaces of Constant Curvature, 5th Ed., Publish or Perish, Delaware 1984. MR 88k:53002 [Z] S. Zhu. The Comparison Geometry of Ricci Curvature,preprint.

Department of Mathematics, Pennsylvania State University, Altoona, Pennsylvania 16601 E-mail address: [email protected]

License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use