Pseudo-Isometric Surgery

TL;DR

This paper introduces a new metric space surgery method called pseudo-isometric surgery, analyzing how properties like pseudo-isometry are preserved during the process, with implications for geometric group theory.

Contribution

It defines a novel surgery on metric spaces and investigates the inheritance of pseudo-isometry properties, highlighting limitations for quasi-isometries.

Findings

Pseudo-isometry is preserved under the surgery.

Quasi-isometry is not necessarily preserved.

Provides a framework for modifying metric spaces while controlling certain properties.

Abstract

We introduce a type of surgery on metric spaces. This surgery, in some sense, seeks to replace a subspace $S$ of a metric space $X$ with another metric space $T$ via a function $f : S \to T$. When $T$ is a discrete space, this amounts to collapsing the subspace according to the function. This surgery results in a new metric space we denote $\widehat{X}_f$ and there is a natural function $F : X \to \widehat{X}_f$ induced from $f$. Our primary interest is investigating if properties of the original function $f$ are inherited by the induced function $F$. We show that if $f$ is a pseudo-isometry then so is $F$. However, for a quasi-isometry, a very natural generalization of a pseudo-isometry that is prevalent in geometric group theory, such a result does not hold.