Exact boundedness exponents for permutation and automorphism groups

We determine exact boundedness exponents for automorphism groups in the pointwise topology. For countably infinite-dimensional vector spaces over countable fields, general linear groups have least open-exhaustion exponent three over finite fields and four over infinite fields. Symplectic groups for the standard alternating form have corresponding exponents three and five. Orthogonal groups for the form $B(x,y)=\sum_i x_i y_i$ have the same three-versus-five dichotomy in characteristic different from two. The automorphism group of a free abelian group of countably infinite rank has least open-exhaustion exponent five. Routing and reflection give exact exponent-four results for unrestricted permutation wreath products. A directional fresh-copy criterion and function-group applications supplement these calculations.

This paper develops material from the author's monograph Property (OB) and the Coarse Geometry of Polish Groups. It is one of eight separate papers intended to replace the monograph.

GitHub repository: https://github.com/eulogica/property-ob-independent-papers