##
**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.

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.

Reviewer: Joan Porti (Bellaterra)

### MSC:

57N10 | Topology of general \(3\)-manifolds (MSC2010) |

30F40 | Kleinian groups (aspects of compact Riemann surfaces and uniformization) |

20E07 | Subgroup theorems; subgroup growth |

### Keywords:

hyperbolic 3-orbifold; tower of covering spaces; mod \(p\) homology; Heegaard gradient; large group; property tau### Citations:

Zbl 1110.57015### References:

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