Logarithmically Slim Polygons
Abstract
We study the relationship between the large-scale geometry of a geodesic metric space and the slimness of its geodesic polygons. We show that if there is a uniform constant δ such that every geodesic polygon in an unbounded geodesic metric space is δ-slim, then the space has asymptotic Assouad--Nagata dimension one. We further consider graphs of bounded valency and allow the slimness constant of an n-gon to grow logarithmically with n. More precisely, under an explicit logarithmic bound on the slimness of geodesic n-gons, we prove that the graph has asymptotic Assouad--Nagata dimension one. Thus, sufficiently strong logarithmic control on the slimness of geodesic polygons forces one-dimensional large-scale geometry. We also obtain two criteria for graphs, outside the hyperbolic setting, to have asymptotic Assouad--Nagata dimension one. In particular, these criteria show that one-dimensional large-scale geometry can be detected from quantitative restrictions on the slimness of geodesic polygons, without requiring the space to be hyperbolic.
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BibTeX
@misc{tselekidis2026logarithmically,
title = {{Logarithmically Slim Polygons}},
author = {Panagiotis Tselekidis},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.05429},
eprint = {2611.05429},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.05429},
}
RIS
TY - UNPB
TI - Logarithmically Slim Polygons
AU - Panagiotis Tselekidis
PY - 2026
DA - 2026/11/03
PB - arXiv
DO - 10.48550/arxiv.2611.05429
UR - https://arxiv.org/abs/2611.05429
AB - We study the relationship between the large-scale geometry of a geodesic metric space and the slimness of its geodesic polygons. We show that if there is a uniform constant δ such that every geodesic polygon in an unbounded geodesic metric space is δ-slim, then the space has asymptotic Assouad--Nagata dimension one. We further consider graphs of bounded valency and allow the slimness constant of an n-gon to grow logarithmically with n. More precisely, under an explicit logarithmic bound on the slimness of geodesic n-gons, we prove that the graph has asymptotic Assouad--Nagata dimension one. Thus, sufficiently strong logarithmic control on the slimness of geodesic polygons forces one-dimensional large-scale geometry. We also obtain two criteria for graphs, outside the hyperbolic setting, to have asymptotic Assouad--Nagata dimension one. In particular, these criteria show that one-dimensional large-scale geometry can be detected from quantitative restrictions on the slimness of geodesic polygons, without requiring the space to be hyperbolic.
ER -