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Geometry of outer spaces and outer automorphism groups

16–20 April 2018

Organisers: Ruth Charney, Karen Vogtmann

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Groups of automorphisms of free groups have been studied since the beginning of group theory by both algebraic and topological methods. There has recently been a great deal of progress on understanding these groups, driven by analogies with arithmetic groups and mapping class groups. In particular Culler and Vogtmann introduced a space called ''Outer space" which plays for $Out(F_n)$ the role symmetric spaces play for arithmetic groups and and Teichmüller space plays for mapping class groups. This has inspired the introduction of other tools analogous to those used for arithmetic and mapping class groups, including the free-splitting and free-factor complexes (as analogues of the Tits building and curve complex) and Bestvina-Handel's ''train-track" technology for automorphisms of free groups (inspired by Thurston's notion of train tracks for surface automorphisms).

During this workshop we will both develop these analogies further and work on extending them to the outer automorphism groups of a larger family of groups, namely right-angled Artin groups (RAAGs). Geometric aspects of Outer space, using its natural (asymmetric) Lipschitz metric, are in early stages of development. Geometric problems inspired by the analogies with arithmetic and mapping class groups include finding distance formulae in the free-splitting or free-factor complexes that approximate distances in Outer space and finding purely fully-irreducible surface subgroups of $Out(F_n)$. Computational problems include finding efficient train track algorithms and computing topological invariants such as cohomology. For right-angled Artin groups, initial steps in this programme have been taken by the very recent introduction of an ''Outer space for RAAGs" which combines features of Culler-Vogtmann space and the symmetric space $SL(n,\Bbb{R})/SO(n)$, as might be expected from the fact that RAAGs interpolate between free groups and free abelian groups. Understanding fundamental properties of this space will give new insight into automorphism groups of RAAGs.


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Mathematical Interdisciplinary Research at Warwick (MIR@W)
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