# American Institute of Mathematical Sciences

doi: 10.3934/dcds.2020320

## Gromov-Hausdorff stability for group actions

 1 Department of Mathematics, College of Science, Yanbian University, No. 977, Gongyuan Road, Yanji City 133002, Jilin Province, China 2 Department of Mathematics, Chungnam National University, Daejeon 305-764, Korea 3 Instituto de Matemática, Universidade Federal do Rio de Janeiro, P. O. Box 68530 21945-970, Rio de Janeiro, Brazil

Received  February 2020 Revised  July 2020 Published  August 2020

Fund Project: KL and MD were supported by the NRF grant funded by the Korea government (MSIT)(NRF-2018R1A2B3001457). CAM was partially supported by the NRF Brain Pool Grant funded by the Korea government and CNPq from Brazil

We will extend the topological Gromov-Hausdorff stability [2] from homeomorphisms to finitely generated actions. We prove that if an action is expansive and has the shadowing property, then it is topologically GH-stable. From this we derive examples of topologically GH-stable actions of the discrete Heisenberg group on tori. Finally, we prove that the topological GH-stability is an invariant under isometric conjugacy.

Citation: Meihua Dong, Keonhee Lee, Carlos Morales. Gromov-Hausdorff stability for group actions. Discrete & Continuous Dynamical Systems - A, doi: 10.3934/dcds.2020320
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