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Abstract Suppose that is a free product , where each of the groups is torsion‐free and is a free group of rank . Let be the deformation space associated to this free product decomposition. We show that the diameter of the projection of the subset of where a given element has bounded length to the ‐factor graph is bounded, where the diameter bound depends only on the length bound. This relies on an analysis of the boundary of as a hyperbolic group relative to the collection of subgroups together with a given nonperipheral cyclic subgroup. The main theorem is new even in the case that , in which case is the Culler–Vogtmann outer space. In a future paper, we will apply this theorem to study the geometry of free group extensions.
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Subgroups of bounded rank in hyperbolic 3‐manifold groups
Abstract We prove a finiteness theorem for subgroups of bounded rank in hyperbolic 3‐manifold groups. As a consequence, we show that every bounded rank covering tower of closed hyperbolic 3‐manifolds is a tower of finite covers associated to a fibration over a 1‐orbifold.
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- PAR ID:
- 10549796
- Publisher / Repository:
- Oxford University Press (OUP)
- Date Published:
- Journal Name:
- Bulletin of the London Mathematical Society
- Volume:
- 56
- Issue:
- 12
- ISSN:
- 0024-6093
- Format(s):
- Medium: X Size: p. 3829-3837
- Size(s):
- p. 3829-3837
- Sponsoring Org:
- National Science Foundation
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