# Jerusalem Mathematics Colloquium

יום חמישי, י'ב בסיון, תשס"ג

Thursday, 12th June 2003, 4:00 pm

Mathematics Building, Lecture Hall 2

##

Professor Mark Sapir

(Vanderbilt)

"Diagram groups of directed 2-complexes, spaces of positive paths, homology
and homotopy"

** Abstract: ** Diagram groups are fundamental groups of spaces of
positive paths of directed 2-complexes. These spaces are K(.,1) and their
universal covers are CAT(0) cubical complexes. In particular, the R. Thompson
group F is the diagram group of the Dunce hat. This fact allows us to find
cenralizers of elements in F, solve conjugacy problem there, etc.

We show that universal covers of spaces of positive paths are themselves
spaces of positive paths of the so called rooted 2-trees. Using that, we
prove that all homology groups of diagram groups are free abelian, and
that the diagram groups are bi-orderable. We construct many new examples
of infinite dimensional FP_infty groups with rational Poincare series, show
that basically "any" rational function can be the Poincare function of
an FP_infty group. We also construct an FP_infty diagram group containing
all countable diagram groups.

I am going to talk also about representations of diagram groups by
homeomorphisms of the real line, and about rigidity results involving diagram
groups. The latter results are similar in spirit to the flat torus theorem
but the groups involved are more complicated.

This is joint work with Victor Guba.

Light refreshments will be served in the faculty lounge at 3:30.

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