Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · math/0412219

Roundness properties of groups

J. -F. Lafont·S. Prassidis
View original source ↗Read source document ↗Export metadata ↓

Abstract

Roundness of metric spaces was introduced by Per Enflo as a tool to study uniform structures of linear topological spaces. The present paper investigates geometric and topological properties detected by the roundness of general metric spaces. In particular, we show that geodesic spaces of roundness 2 are contractible, and that a compact Riemannian manifold with roundness $>1$ must be simply connected. We then focus our investigation on Cayley graphs of finitely generated groups. One of our main results is that every Cayley graph of a free abelian group on $\geq 2$ generators has roundness $=1$. We show that if a group has no Cayley graph of roundness $=1$, then it must be a torsion group with every element of order $2,3,5$, or 7.

Explore related subjects

math.MGmath.GR

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

J. -F. Lafont, S. Prassidis. 2004-12-10. Roundness properties of groups. https://arxiv.org/abs/math/0412219

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Many Antipodal Pairs Force Many Neighboring Pairs

Let $X=\{x_1,\dots,x_n\}\subset \mathbb{R}^2$ be a finite set of points of diameter at most $1$. It is natural to expect that if many pairs $(x_i,x_j)$ lie at distance close to $1$ from each other, then some clustering phenomenon must occur, implying that a significant number of these pairs are also very close to each other. %For $0<\varepsilon<1$, we call a pair $(x_i,x_j)$ $\varepsilon$-antipodal if $\|x_i-x_j\|\ge 1-\varepsilon$ and $\varepsilon$-neighboring if $\|x_i-x_j\|\le \varepsilon$. We prove that there exists a universal constant $c>0$ such that for all $0<\varepsilon<1$, whenever $n$ is large enough, we have: \[ \big|\{(i,j):\|x_i-x_j\|\le \varepsilon\}\big| \geq c\cdot \varepsilon^{1/2}\cdot \big|\{(i,j):\|x_i-x_j\|\geq 1-\varepsilon\}\big|. \] This confirms a recent conjecture of Steinerberger, who asked whether the $\varepsilon^{1/2}$ ratio is the best possible. We also study a two-parameter version of Steinerberger's question by considering the number of pairs at distance at most $\varepsilon_1$ and at distance at least $1-\varepsilon_2$. We show that in this case the optimal ratio is $\varepsilon_1^2\cdot\varepsilon_2^{-3/2}$ when $\varepsilon_1\leq\varepsilon_2$, and $\varepsilon_1^{1/2}$ otherwise. The proof proceeds by introducing an auxiliary graph associated with the set $X$ and reducing the problem to bounding the largest eigenvalue of its adjacency matrix. Our main result is the outcome of human--AI interactions using ChatGPT 5.4.

math.MG↗
arXiv2026

Monge's transport problem in synthetic Lorentzian spacetimes

We prove the existence of solutions to the Lorentzian Monge problem on synthetic Lorentzian spacetimes that satisfy the forward timelike measure contraction property $TMCP^+(K,N)$. The cost for this problem is the time separation function $\ell(x,y)$ (the Lorentz distance), which represents the maximum amount a particle can age when traveling from $x$ to $y$. The solution is based on the needle decomposition technique, and amounts to a reduction of the full problem to its one-dimensional counterparts. As an application of needle decomposition, we introduce the blended $TMCP$ condition, and obtain a quantitative refinement of the synthetic Hawking singularity theorem. We additionally prove a timelike nonbranching reversibility theorem for $TCD_p(K,N)$ spacetimes, showing that forward $p$-essentially nonbranching spacetimes are backward $p$-essentially nonbranching.

math.MG↗
arXiv2026

Metric foundations of geometry

A metric space is called all-set-homogeneous if every isometry between two of its subsets extends to an isometry of the whole space. We classify all-set-homogeneous length spaces: besides the classical examples, they include the universal metric trees of finite valence.

math.MG↗