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Automorphisms of free groups and the moduli space of graphs


The objective is to study the automorphism groups of the free group using methods from homotopy theory. We hope to obtain new homological information about these groups. The classifying space of the automorphism groups of free groups are models formoduli spaces of graphs. On the one hand, one can use graphs to study the groups. On the other hand, one should obtain information about graphs by studying those groups. Hather's homology stability result leads us to study the stable automorphism group. We conjecture that its classifying space has the homotopy type of the classifying space of the infinite symmetric group, after plus-construction. Recently, progress w made in the study of the mapping class groups. Those groups, which I studied in my thesis, have properties similar to the properties of the automorphism groups. There is a map from the mapping class groups to the automorphism groups. Asit is now known that both stable groups are infinite loop spaces, we hope to show, as a first step in this program, that this map is an infinite loop map. This should enable us to transport our newly gained knowledge of the mapping class groups to the automorphism groups. Ib Madsen is a leading expert in the field of homotopy theory. Moreover, I would be joining a group of people in Aarhus working on questions similar to the ones I want to approach. Although I would now be considered to be a homotopy theorist, I started my career in combinatorics ( my first publication was in that field). As we use combinatorial models for the large spaces we deal with in homotopy theory, I hope my skills will bring something to the group.


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