Plastic metric spaces and groups

TL;DR

This paper investigates plastic metric spaces and groups, introducing three main results: (i) no countable dense subspace of a normed space is plastic, (ii) every $k$-crowded separable metric space contains a plastic dense subspace, and (iii) every strictly convex separable metric group contains a plastically rigid dense subgroup. The authors develop a multi-pronged approach combining LCDH theory, Cantor-set techniques in analytic spaces, and a transfinite inductive construction to produce dense plastically rigid subgroups in strictly convex groups. They prove that countable dense subsets fail to be plastic via completions and contractive bijections, construct plastic dense subspaces in $k$-crowded spaces, and build plastically rigid dense subgroups in strictly convex metric abelian groups. The work highlights rigidity phenomena for non-expansive bijections and poses several set-theoretic and geometric open problems, including questions about analytic density and plasticity under different axioms and norms.

Abstract

A metric space is plastic if all its non-expansive bijections are isometries. We prove three main results: (1) every countable dense subspace of a normed space is not plastic, (2) every $k$-crowded separable metric space contains a plastic dense subspace, and (3) every strictly convex separable metric group contains a plastic dense subgroup.

Theorems & Definitions (40)

• Theorem 1
• Theorem 2
• Theorem 3
• Definition 1
• Definition 2
• Lemma 1
• proof
• Proposition 1
• Definition 3
• Example 1