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We are interested in the structure of the set of homomorphisms from a fixed (but arbitrary) finitely generated group G to the groups in some fixed family (such as the family of 3-manifold groups). I will explain what one might hope to say in different situations, and explain some applications to relatively hyperbolic groups and acylindrically hyperbolic groups, and some hoped-for applications to 3-manifold groups.
This is joint work with Michael Hull and joint work in preparation with Michael Hull and Hao Liang.
We are interested in the structure of the set of homomorphisms from a fixed (but arbitrary) finitely generated group G to the groups in some fixed family (such as the family of 3-manifold groups). I will explain what one might hope to say in different situations, and explain some applications to relatively hyperbolic groups and acylindrically hyperbolic groups, and some hoped-for applications to 3-manifold groups.
This is joint work with Michael ...

57N10 ; 20F65 ; 20F67 ; 20E08 ; 57M07

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Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of Gleason and Montgomery-Zippin) that it suffices to rule out the case of the additive group of p-adic integers acting faithfully on a manifold. I will present a solution in dimension three.
Hilbert's Fifth Problem asks whether every topological group which is a manifold is in fact a (smooth!) Lie group; this was solved in the affirmative by Gleason and Montgomery-Zippin. A stronger conjecture is that a locally compact topological group which acts faithfully on a manifold must be a Lie group. This is the Hilbert--Smith Conjecture, which in full generality is still wide open. It is known, however (as a corollary to the work of ...

57N10

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