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4:00 pm Monday, September 25, 2017 Topology Seminar: Surface presentations of groups.by John Hempel (Rice) in HBH 227- Every finite presentation P of a group G can be represented by two sets X,Y of pairwise disjoint, oriented simple closed curves on a closed oriented surface, S: generators correspond to components of X and relators to components of Y – which are written as words in the generators by their intersection patterns. This data also determines a compact pseudo (possibly real) 3-manifold M∗ and we observe that G is isomorphic to π_1(M^∗) ∗ F_{β_0(S−X)−1}. In particular every finitely presented group is the fundamental group of some pseudo 3-manifold. The free factor F_{β_0(S−X)−1} can be easily delt with and we see that the minimal genus among all such surfaces S gives an invariant g(P) which measures ”how far” M^∗ is from a 3-manifold. For G freely indecomposable M^∗ is a 3-manifold if and only if g(P) = β_0(X). Cognizant of the fact that simply defined invariants may turn out to be pretty much useless, I decided to see if I could compute any interesting examples before I publicized these ideas. I have done this for (standard presentations) of finitely generated Abelian groups, and will discuss these results.
Submitted by neil.fullarton@rice.edu |