Pseudo-Isometric Surgery
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866909714010341376 |
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| author | Clay, Matt Thompson, Josh |
| author_facet | Clay, Matt Thompson, Josh |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23666 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Pseudo-Isometric Surgery Clay, Matt Thompson, Josh Metric Geometry 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. |
| title | Pseudo-Isometric Surgery |
| topic | Metric Geometry |
| url | https://arxiv.org/abs/2507.23666 |