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Covering spaces of 3-orbifolds
Geometric Topology · Submitted 1 Aug 2005
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime p.
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The abstract, sentence by sentence
Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime p. This means that the dimension of the first homology, with mod p coefficients, of the fundamental group of O_i grows linearly in the covering degree. The proof combines techniques from 3-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about p-adic analytic pro-p groups. This has several consequences. Firstly, the fundamental group of O has at least exponential subgroup growth. Secondly, the covers {O_i} have positive Heegaard gradient. Thirdly, we use it to show that a group-theoretic conjecture of Lubotzky and Zelmanov would imply that O has large fundamental group. This implication uses a new theorem of the author, which will appear in a forthcoming paper. These results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3-orbifold with non-empty singular locus has large fundamental group. Many of the above results apply also to 3-manifolds commensurable with an orientable finite-volume hyperbolic 3-orbifold with non-empty singular locus. This includes all closed orientable hyperbolic 3-manifolds with rank two fundamental group, and all arithmetic 3-manifolds.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. | background 0.96 | 0.09 | 0.02 | asserted 0.86 |
| s1 | (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime p. | result 0.94 | 0.93 | 0.87 | asserted 1.00 |
| s2 | This means that the dimension of the first homology, with mod p coefficients, of the fundamental group of O_i grows linearly in the covering degree. | interpretation 0.93 | 0.52 | 0.13 | asserted 0.99 |
| s3 | The proof combines techniques from 3-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about p-adic analytic pro-p groups. | method 1.00 | 0.54 | 0.02 | asserted 0.99 |
| s4 | This has several consequences. | interpretation 0.82 | 0.40 | 0.02 | asserted 0.61 |
| s5 | Firstly, the fundamental group of O has at least exponential subgroup growth. | result 0.84 | 0.84 | 0.08 | asserted 1.00 |
| s6 | Secondly, the covers {O_i} have positive Heegaard gradient. | result 0.99 | 0.90 | 0.05 | asserted 1.00 |
| s7 | Thirdly, we use it to show that a group-theoretic conjecture of Lubotzky and Zelmanov would imply that O has large fundamental group. | interpretation 0.56 | 0.86 | 0.07 | asserted 0.94 |
| s8 | This implication uses a new theorem of the author, which will appear in a forthcoming paper. | future work 0.59 | 0.80 | 0.02 | asserted 0.98 |
| s9 | These results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3-orbifold with non-empty singular locus has large fundamental group. | interpretation 1.00 | 0.40 | 0.33 | hedged possibility 0.87 |
| s10 | Many of the above results apply also to 3-manifolds commensurable with an orientable finite-volume hyperbolic 3-orbifold with non-empty singular locus. | result 0.85 | 0.76 | 0.04 | asserted 0.79 |
| s11 | This includes all closed orientable hyperbolic 3-manifolds with rank two fundamental group, and all arithmetic 3-manifolds. | result 0.51 | 0.53 | 0.03 | asserted 1.00 |
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The paper
Original abstract
Let O be a compact orientable 3-orbifold with non-empty singular locus and a finite volume hyperbolic structure. (Equivalently, O is the quotient of hyperbolic 3-space by a lattice in PSL(2,C) with torsion.) Then we prove that O has a tower of finite-sheeted covers {O_i} with linear growth of p-homology, for some prime p. This means that the dimension of the first homology, with mod p coefficients, of the fundamental group of O_i grows linearly in the covering degree. The proof combines techniques from 3-manifold theory with group-theoretic methods, including the Golod-Shafarevich inequality and results about p-adic analytic pro-p groups. This has several consequences. Firstly, the fundamental group of O has at least exponential subgroup growth. Secondly, the covers {O_i} have positive Heegaard gradient. Thirdly, we use it to show that a group-theoretic conjecture of Lubotzky and Zelmanov would imply that O has large fundamental group. This implication uses a new theorem of the author, which will appear in a forthcoming paper. These results all provide strong evidence for the conjecture that any closed orientable hyperbolic 3-orbifold with non-empty singular locus has large fundamental group. Many of the above results apply also to 3-manifolds commensurable with an orientable finite-volume hyperbolic 3-orbifold with non-empty singular locus. This includes all closed orientable hyperbolic 3-manifolds with rank two fundamental group, and all arithmetic 3-manifolds.
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