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Thursday, October 5, 2017

4:20 PM - 5:20 PM (ET)

Exley Science Center Tower ESC 121

Event Type

Seminar/Colloquium

Contact

Canalia, Caryn

2182

Link

https://eaglet.wesleyan.edu/MasterCalendar/EventDetails.aspx?EventDetailId=79075

Speaker: Shelly Harvey, Rice University

Title: Corank of 3-manifold groups G with H_2(G)=0

** Abstract:
**The corank of a group G, c(G), is the maximal r such that there is a
surjective homomorphism from G to a non-abelian free group of rank r. We
note that for any group G, c(G) is bounded above by b_1(G), the rank of the
abelianization of G. For closed surface groups S, we have a further
relationship between these two complexities, namely b_1(S) = 2 c(S). It was
asked whether such a relationship exists for 3-manifold groups. In a
previous paper, I showed that there were closed 3-manifold groups G with b_1(G)
arbitrarily large but with c(G)=1. It was asked by Michael Freedman
whether such a statement was known when the group was the group of a 3-dimensional
homology handlebody. These groups are much more subtle and have
properties that make them look like a free group so the question becomes much
more difficult. In fact, all of the
previous techniques used by the author fail. The complete answer to the question
is still unknown. However, we show that there are groups G_m (for all m
\geq 2) which are the fundamental group of a 3-dimensional handlebody (in
particular, H_2(G_m)=0) and satisfy the following: b_1(G_m)= m and c(G_m)=f(m)
where f(m)=m/2 for m even and f(m)=(m+1)/2 for m odd. This is joint work
with Eamonn Tweedy.