The recent proof of Thurston's virtual fibering conjecture brought together tools at the forefront of geometric group theory, dynamics, and hyperbolic geometry. We still lack, however, an effective or constructive understanding of three-dimensional hyperbolic geometry, and more generally, 3-manifold topology. For example, a closed hyperbolic 3-manifold admits a finite cover which fibers over the circle, but can one construct such a cover from a presentation of the fundamental group? Can one implement an algorithm -- perhaps with the help of preexisting software such as SnapPea -- to obtain such a cover? While much work remains, both computation and theory have progressed. Fast algorithms have been developed for running computations in the mapping class group and other finitely generated groups, as well as for recognizing certain types 3-manifolds and knot and link complements up to homeomorphism. These have been supplemented by a new wave of constructive theorems which explicitly relate the algebra of the fundamental group of a hyperbolic 3-manifold to its geometry, and to the geometry of various simplicial complexes, such as the curve complex. This ICERM workshop will focus on such advances, as well as on the development of new algorithms and extension of algorithmic techniques to the study of free groups. The workshop aims to bring together researchers from a broad range of related fields to work towards a more effective and quantitative understanding of 3-manifold topology, geometric group theory, and hyperbolic geometry.
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