A local-to-global fixed point theorem for CAT(0) spaces with a uniform elliptic gap
Abstract
A finitely generated group acting locally elliptically on a complete \CAT(0) space of finite geometric dimension fixes a point if distances to elliptic fixed sets are uniformly controlled by displacements. For arbitrary groups, the same hypothesis gives a fixed point in the space or its visual boundary. The proof combines a first-moment random-walk displacement estimate with Izeki's zero-drift theorem. For finite-rank R-buildings whose Euclidean metric factor has finite linear image, we obtain an all-type local-to-global theorem, including the nondiscrete G̃2 case, with an explicit gap and without metric completeness; the linear-image restriction can be removed from the fixed-point conclusion when the Euclidean factor has dimension at most three. A tangent-cone criterion characterizes the gap and yields a cubical bound with optimal asymptotic growth d/π in dimension d. For Hadamard manifolds, the gap is equivalent to uniformly bounded finite stabilizers in the isometry image, with stabilizer bounds of optimal polynomial growth in the gap constant.
Related
- A Fractional Logistic-Type Elliptic Problem with Global Interactions: Existence, Local Uniqueness, and the Fractional-to-Local Limit
- Local-to-Global Isometries in Standard-Norm Generalized Gyrovector Spaces: Thompson and Hilbert Geometries of JB-Cones
- Uniform estimates for elliptic systems with critical Sobolev coefficients
- From qualitative smoothness to uniform estimates for fully nonlinear elliptic equations
- Frey-Mazur theorem for semistable elliptic curves over function fields
BibTeX
@misc{schillewaert2026localtoglobal,
title = {{A local-to-global fixed point theorem for $\CAT(0)$ spaces with a uniform elliptic gap}},
author = {Jeroen Schillewaert},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.41468},
eprint = {2610.41468},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.41468},
}
RIS
TY - UNPB
TI - A local-to-global fixed point theorem for CAT(0) spaces with a uniform elliptic gap
AU - Jeroen Schillewaert
PY - 2026
DA - 2026/10/30
PB - arXiv
DO - 10.48550/arxiv.2610.41468
UR - https://arxiv.org/abs/2610.41468
AB - A finitely generated group acting locally elliptically on a complete \CAT(0) space of finite geometric dimension fixes a point if distances to elliptic fixed sets are uniformly controlled by displacements. For arbitrary groups, the same hypothesis gives a fixed point in the space or its visual boundary. The proof combines a first-moment random-walk displacement estimate with Izeki's zero-drift theorem. For finite-rank R-buildings whose Euclidean metric factor has finite linear image, we obtain an all-type local-to-global theorem, including the nondiscrete G̃_2 case, with an explicit gap and without metric completeness; the linear-image restriction can be removed from the fixed-point conclusion when the Euclidean factor has dimension at most three. A tangent-cone criterion characterizes the gap and yields a cubical bound with optimal asymptotic growth d/π in dimension d. For Hadamard manifolds, the gap is equivalent to uniformly bounded finite stabilizers in the isometry image, with stabilizer bounds of optimal polynomial growth in the gap constant.
ER -