Seminar

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Let M be a compact, orientable, hyperbolizable 3-manifold with non-empty boundary, and let ρi be a sequence of discrete, non-faithful representations of π1(M). We say that ρi converges strongly if its algebraic limit agrees with its geometric limit. The limit of strong convergence is known to preserve many nice properties of the approximating sequence, including the homeomorphism types and critical exponent (if M is not a handlebody).



The conditions ensuring strong convergence were studied by Anderson and Canary for the faithful representations, and later extended by Biringer and Souto to the non-faithful setting. In particular, Biringer and Souto showed that, in the absence of parabolic elements in the algebraic limit, strong convergence depends on whether all degenerate ends in the algebraic limit embed in the geometric limit. They also constructed multiple non-strongly convergent examples for surface groups and compression bodies, in which the algebraic limit has a non-embedded degenerate end.



In this talk, I will build on these results and discuss families of 3-manifolds for which all degenerate ends always embed. I will also show that the non-embedding phenomena appearing in the known examples are not limited to surface groups and compression bodies, but exist more generally in families of 3-manifold groups.