Sublinearly Morse boundaries from the viewpoint of combinatorics

Merlin Incerti-Medici; Abdul Zalloum

doi:10.1515/forum-2022-0269

Article

Sublinearly Morse boundaries from the viewpoint of combinatorics

Merlin Incerti-Medici and Abdul Zalloum

Published/Copyright: March 31, 2023

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From the journal Forum Mathematicum Volume 35 Issue 4

Abstract

We prove that the sublinearly Morse boundary of CAT ⁢ ( 0 ) cubulated groups with factor systems continuously injects in the Gromov boundary of a certain hyperbolic graph Γ. We also show that for all CAT ⁢ ( 0 ) cube complexes, convergence to sublinearly Morse geodesic rays has a simple combinatorial description using the hyperplanes crossed by such sequences. As an application of this combinatorial description, we show that a certain subspace of the Roller boundary continuously surjects on the subspace of the visual boundary consisting of sublinearly Morse geodesic rays.

Keywords: Geometric group theory; (sublinearly) Morse boundaries; asymptotic geometry

MSC 2020: 20F65

Communicated by Clara Löh

Acknowledgements

The authors would like to thank Carolyn Abbott, Ruth Charney, Matthew Durham, Elia Fioravanti, Anthony Genevois, Mark Hagen, Qing Liu, Kasra Rafi, and Jacob Russell for fruitful discussions. Finally, we would like to thank the referee whose comments helped improve the exposition considerably.

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Received: 2022-09-14

Revised: 2023-01-08

Published Online: 2023-03-31

Published in Print: 2023-07-01

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Articles in the same Issue

https://doi.org/10.1515/forum-2022-0269

Keywords for this article

Geometric group theory; (sublinearly) Morse boundaries; asymptotic geometry