## Covering spaces of 3-orbifolds.(English)Zbl 1109.57015

In this interesting paper the author studies towers of finite sheeted coverings of hyperbolic three-orbifolds with finite volume and non-empty singular locus. It can be seen as a continuation in the orbifold setting of a previous work of the author [Invent. Math. 164, No. 2, 317–359 (2006; Zbl 1110.57015)], using the non-empty singular locus as a tool to study the covers.
The main result is the existence of a tower of finite sheeted coverings with several properties. Firstly, the rank of the first mod $$p$$ homology group grows linearly, provided that $$p$$ is a prime that divides any ramification index of the orbifold. The singular locus of the coverings can be chosen to be either empty (i.e. a covering by manifolds) or not. In addition, the successive coverings of the tower have degree $$p$$ and are regular.
This result has several consequences. Among others, it is shown that the fundamental group of the orbifold has at least exponential subgroup growth. Namely, the number of subgroups of index at most $$n$$ grows at least exponentially with $$n$$. It is also shown that, if a conjecture of Lubotzky and Zelmanov holds true, then the fundamental group is large, which means that a finite index subgroup has a surjection onto a free group of rank two.
Those results apply to closed hyperbolic manifolds commensurable to orbifolds with non-empty singular locus. In particular to arithmetic hyperbolic three-manifolds, according to a preprint of M. Lackenby, D. D. Long and A. W. Reid [Covering spaces of arithmetic 3-orbifolds; arXiv:math.GT/0601677].
As indicated in the abstract, the proof combines techniques of three-manifold theory with group-theoretical methods, including the Golod-Shafarevich inequality and results about $$p$$-adic pro-$$p$$ groups.

### MSC:

 57N10 Topology of general $$3$$-manifolds (MSC2010) 30F40 Kleinian groups (aspects of compact Riemann surfaces and uniformization) 20E07 Subgroup theorems; subgroup growth

Zbl 1110.57015
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### References:

 [1] M. Boileau, B. Leeb, and J. Porti, Geometrization of,$$3$$-dimensional orbifolds , Ann. of Math. (2) 162 (2005), 195–290. · Zbl 1087.57009 [2] J. Button, Strong Tits alternatives for compact $$3$$-manifolds with boundary, J. Pure Appl. Algebra 191 (2004), 89–98. · Zbl 1056.57001 [3] D. Cooper, C. D. Hodgson, and S. P. Kerckhoff, Three-Dimensional Orbifolds and Cone-Manifolds , MSJ Mem. 5 , Math. Soc. Japan, Tokyo, 2000. · Zbl 0955.57014 [4] D. Cooper, D. D. Long, and A. W. Reid, Essential closed surfaces in bounded $$3$$-manifolds, J. Amer. Math. Soc. 10 (1997), 553–563. JSTOR: · Zbl 0896.57009 [5] D. Gabai, G. R. Meyerhoff, and N. Thurston, Homotopy hyperbolic $$3$$-manifolds are hyperbolic , Ann. of Math. (2) 157 (2003), 335–431. JSTOR: · Zbl 1052.57019 [6] M. Lackenby, Heegaard splittings, the virtually Haken conjecture and property $$(\tau),$$ Invent. Math. 164 (2006), 317–359. · Zbl 1110.57015 [7] -, Large groups, property $$(\tau)$$ and the homology growth of subgroups , · Zbl 1185.57014 [8] -, Some $$3$$-manifolds and $$3$$-orbifolds with large fundamental group , · Zbl 1129.57025 [9] M. Lackenby, D. D. Long, and A. W. Reid, Covering spaces of arithmetic $$3$$-orbifolds, [10] A. Lubotzky, Group presentations, $$p$$-adic analytic groups and lattices in $$\mathrm SL_2(C)$$ Ann. of Math. (2) 118 (1983), 115–130. JSTOR: · Zbl 0541.20020 [11] -, “Dimension function for discrete groups” in Proceedings of,Groups (St. Andrews , Scotland, 1985) , London Math. Soc. Lecture Note Ser. 121 , Cambridge Univ. Press, Cambridge, 1986, 254–262. [12] -, Discrete Groups, Expanding Graphs and Invariant Measures , Progr. Math. 125 , Birkhäuser, Basel, 1994. [13] -, Eigenvalues of the Laplacian, the first Betti number and the congruence subgroup problem, Ann. of Math. (2) 144 (1996), 441–452. JSTOR: · Zbl 0885.11037 [14] -, private communication, 2005. [15] A. Lubotzky and A. Mann, Powerful $$p$$-groups, II: $$p$$-adic analytic groups, J. Algebra 105 (1987), 506–515. · Zbl 0626.20022 [16] A. Lubotzky and D. Segal, Subgroup Growth , Progr. Math. 212 , Birkhäuser, Basel, 2003. [17] J. G. Ratcliffe, “Euler characteristics of $$3$$-manifold groups and discrete subgroups of $$\mathrm SL(2, \mathbb C)$$” in Proceedings of the Northwestern Conference on Cohomology of Groups (Evanston, Ill., 1985) , J. Pure Appl. Algebra 44 , Elsevier, Amsterdam, 1987, 303–314. · Zbl 0627.57003 [18] P. Scott, The geometries of $$3$$-manifolds, Bull. London Math. Soc. 15 (1983), 401–487. · Zbl 0561.57001 [19] W. P. Thurston, The geometry and topology of $$3$$-manifolds , lecture notes, Princeton Univ., Princeton, 1978.
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