For any finite group G, it is easy to see there exists a compact, oriented 3 dimensional manifold M with G as a quotient of the fundamental group of M. However, we can ask more detailed questions about the possible finite quotients of 3-manifold groups, e.g. for G and H finite groups, does there exist a 3-manifold group with G as a quotient but not H as a quotient? We answer all such questions. To prove non-existence, we prove new parity properties of the fundamental groups of 3-manifolds. To prove existence of 3-manifolds with certain finite quotients but not others, we use a probabilistic method, by first proving a formula for the distribution of the fundamental group of a random 3-manifold, in the sense of Dunfield-Thurston. This is joint work with Will Sawin.

Participants are requested to submit their questions for the speaker through the following form: https://forms.gle/WT342H7DonoaRrLU7

The above form will stop accepting responses on **April**** ****18****, 2022** at **18:00 hrs** Indian time.

This colloquium is part of the Distinguished Lecture Series (DLS) organized by The (Indian) Mathematics Consortium (TMC), and co-hosted by IIT-B and ICTS-TIFR. For more information about this lecture series, and to register, please visit the homepage of the TMC DLS.