Pseudo-Isometric Surgery

2025/07/31 by Clay, Matt, Thompson, Josh
#FOS: Mathematics #Metric Geometry (math.MG)

paper · doi:10.48550/arxiv.2507.23666

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 → 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 \widehatX_f and there is a natural function F : X → \widehatX_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.

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