Simply transitive geodesics and omnipotence of lattices in PSL$(2,\mathbb{C})$

Ian Agol; Tam Cheetham-West; Yair Minsky

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arXiv · 2409.08418

Simply transitive geodesics and omnipotence of lattices in PSL$(2,\mathbb{C})$

Ian Agol·Tam Cheetham-West·Yair Minsky

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Abstract

We show that the isometry group of a finite-volume hyperbolic 3-manifold acts simply transitively on many of its closed geodesics. Combining this observation with the Virtual Special Theorems of the first author and Wise, we show that every non-arithmetic lattice in PSL$(2,\mathbb{C})$ is the full group of orientation-preserving isometries for some other lattice and that the orientation-preserving isometry group of a finite-volume hyperbolic 3-manifold acts non-trivially on the homology of some finite-sheeted cover.

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Ian Agol, Tam Cheetham-West, Yair Minsky. 2024-09-12. Simply transitive geodesics and omnipotence of lattices in PSL$(2,\mathbb{C})$. https://arxiv.org/abs/2409.08418

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