The mapping class group of a shift and the Higman-Thompson groups

Feyisayo Olukoya (University of St Andrews)

Monday 13th May 16:00-17:00 Maths 311B

Abstract

The mapping class group of a shift of finite type fits into an algebraic framework of classifying shifts up-to some equivalence: shift-equivalence (topological conjugacy), strong shift equivalence (eventual topological conjugate), flow equivalence and so on. The mapping class group is to flow equivalence what the automorphism group is to shift-equivalence; it is analogous to but different from the mapping class group of surfaces and has connections with Cuntz-Krieger algebras and topological full groups associated to one-sided shifts of finite type. A classification of shifts of finite type up-to flow equivalence is known and depends on the Bowen-Franks group of the defining matrix.

With a narrow focus on the full shift, and with illustrative examples, we will introduce all the necessary background required to define the mapping class group. For this we will follow primarily the articles "Flow equivalence and isotopy for subshifts (Boyle, Carlsen, and Eilers)" and "The mapping class group of a shift of finite type" (Boyle and Chuysurichay).

We then change track and turn to the outer automorphism group out(G_n,r) of the Higman-Thompson groups G_n,r. The Higman-Thompson groups are generalisations, by Higman, of Thompson group V and act by prefix exchanges on the disjoint union of r copies of the cantor space {0,1,...,n}^N. They are the first infinite family of finitely presented (almost) simple groups. Their automorphisms and outer automorphism groups are characterised in the forthcoming article "The further chameleon groups of Richard Thompson and Graham Higman: Automorphisms via dynamics for the Higman-Thompson groups G_n,r. We sketch out a description of these groups and then show that the outer automorphism group of G_n,n-1 is isomorphic to the mapping class group of the full shift over n-symbols. By results in the article "Automorphisms of the generalised Thompson's group T_n,r" it then follows that the outer automorphism group of G_n,r is isomorphic to a normal subgroup of finite index of the mapping class group with an abelian quotient. In particular, out(G_n,1) is isomorphic to the kernel of the Bowen-Franks representation --- a homomorphism from the mapping class group of the full shift onto the group of units of Z_n-1 (the Bowen-Franks group of the associated matrix).

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