Non-reduction of relations in the Gromov space to Polish actions

TL;DR

This paper demonstrates that certain natural equivalence relations in the Gromov space, related to coarse quasi-isometries and Gromov-Hausdorff distance, cannot be simplified to relations arising from Polish group actions.

Contribution

It proves the non-reducibility of specific geometric relations in the Gromov space to Polish group actions, highlighting limitations in classifying these relations.

Findings

Coarse quasi-isometry relation is not reducible to Polish actions.

Gromov-Hausdorff distance relation cannot be reduced to Polish actions.

Establishes boundaries of classification methods for metric space relations.

Abstract

It is shown that, in the Gromov space of isometry classes of pointed proper metric spaces, the equivalence relations defined by existence of coarse quasi-isometries or being at finite Gromov-Hausdorff distance, cannot be reduced to the equivalence relation defined by any Polish action.