Robust fixed points and finite classifying spaces: CAT(0) actions, Banach targets and group fillings

For a finitely presented group, we characterize the fixed-point property for all isometric actions on complete CAT(0) spaces by finite systems of polynomial distance inequalities. An infeasible system gives a uniform tolerance for simultaneous errors in the generating maps, defining relations and midpoint geometry. For finite presentations, this yields an algorithm that halts exactly when the property holds. On genuine CAT(0) spaces and prescribed Banach classes, continuous iterations give locally Hölder retractions onto common fixed sets and continuous choices in parameter families. For one arithmetic presentation we evaluate a nonlinear generator tolerance of 10−232 on genuine CAT(0) spaces.

Starting from OpenAI's group construction, diagram argument and analytic obstruction, we obtain infinite torsion-free hyperbolic groups with finite two-dimensional classifying spaces and the unrestricted CAT(0) fixed-point property. This answers the existence question recorded by Eskenazis, Mendel and Naor (2019); the qualitative conclusion can also be extracted from OpenAI's analytic argument by standard uniformization. The finite certificates and numerical estimates provide its quantitative and arithmetic development. Combined with known spectral and filling constructions, this input gives, for each compact I ⊂ (1, ∞), a two-generated torsion-free hyperbolic group retaining the CAT(0) property and having fixed points for all affine isometric actions on real subquotients of noncommutative Lp spaces, simultaneously for p ∈ I. Using OpenAI's separate non-residual-finiteness construction and Kapovich–Wise, the prescribed-class groups can also have no finite quotients and any sufficiently large second Betti number. In the acylindrically hyperbolic class we obtain finite relative aspherical completions whose second homology differs from that of the prescribed subcomplex by a free abelian summand. The required deterministic diagram argument is proved in an attributed appendix.

GitHub repository: https://github.com/eulogica/uniform-fixed-points-hyperbolic-groups