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DTSTART;TZID=America/Los_Angeles:20180201T160000
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SUMMARY:Mathematics Department Colloquium: Largeness of 3-manifold group that resemble free groupsâ€ť
UID:115140-ucb-events-calendar@berkeley.edu
ORGANIZER;CN="UC Berkeley Calendar Network":
LOCATION:60 Evans Hall
DESCRIPTION:Shelly Harvey\, Rice University\n\nA group is called large if it has a finite index subgroup which surjects onto a non-abelian free group. By work of Agol and Cooper-Long-Reid\, most 3-manifold groups are large\; in particular\, the fundamental groups of hyperbolic 3-manifolds are large. In previous work\, the first author gave examples of closed\, hyperbolic 3-manifolds with arbitrarily large first homology rank but whose fundamental groups do not surject onto a non-abelian free group. We call a group very large if it surjects onto a non-abelian free group. In this paper\, we consider the question of whether the groups of homology handlebodies - which are very close to being free - are very large. We show that the fundamental group of W. Thurston's tripus manifold\, is not very large\; it is known to be weakly parafree by Stallings' Theorem and large by the work of Cooper-Long-Reid since the tripus is a hyperbolic manifold with totally geodesic boundary. It is still unknown if a 3-manifold group that is weakly parafree of rank at least 3 must be very large. In fact\, I know no example of a finitely presented group that is not very large and has H_2(G)=0 and has first homology rank at least 3. More generally we consider the co-rank of the fundamental group\, also known as the cut number of the manifold. For each positive integer g we construct a homology handlebody Y_g of genus g whose group has co-rank equal to r(g)\, where r(g)=g/2 for g even and r(g)=(g+1)/2 for g odd. That is\, these groups are weakly parafree of rank g and surject onto a free group of rank roughly half of g but no larger.\n\nThis work is joint with Eamonn Tweedy.
URL:http://events.berkeley.edu/index.php/calendar/sn/pubaff.html?event_ID=115140&view=preview
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