# Mathematics Colloquium

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.