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A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries

  • Russell Ricks ORCID logo EMAIL logo
Published/Copyright: June 14, 2019
Forum Mathematicum
From the journal Forum Mathematicum Volume 31 Issue 5

Abstract

We prove the following rank rigidity result for proper CAT⁡(0) spaces with one-dimensional Tits boundaries: Let Γ be a group acting properly discontinuously, cocompactly, and by isometries on such a space X. If the Tits diameter of ∂⁡X equals π and Γ does not act minimally on ∂⁡X, then ∂⁡X is a spherical building or a spherical join. If X is also geodesically complete, then X is a Euclidean building, higher rank symmetric space, or a nontrivial product. Much of the proof, which involves finding a Tits-closed convex building-like subset of ∂⁡X, does not require the Tits diameter to be π, and we give an alternate condition that guarantees rigidity when this hypothesis is removed, which is that a certain invariant of the group action be even.

Keywords: CAT(0); dimension; rank rigidity
MSC 2010: 53C20; 53C24; 20F65

Communicated by Anna Wienhard

Funding source: National Science Foundation

Award Identifier / Grant number: NSF 1045119

Funding statement: This material is based upon work supported by the National Science Foundation under grant number NSF 1045119.

Acknowledgements

The author would like to thank Ralf Spatzier for his support and many helpful discussions.

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Received: 2018-05-31
Revised: 2019-04-06
Published Online: 2019-06-14
Published in Print: 2019-09-01

© 2019 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. Selberg integral over local fields
  3. On conjugation orbits of semisimple pairs in rank one
  4. On a topology property for the moduli space of Kapustin–Witten equations
  5. Differentiability of the evolution map and Mackey continuity
  6. A prime geodesic theorem for SL3(ℤ)
  7. Comparison and continuity of Wick-type star products on certain coadjoint orbits
  8. A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank
  9. Linear invariance of intersections on unitary Rapoport–Zink spaces
  10. Maximal subalgebras of finite-dimensional algebras
  11. A universal enveloping algebra for cocommutative rack bialgebras
  12. A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries
  13. Projective objects in the category of pointwise finite dimensional representations of an interval finite quiver
https://doi.org/10.1515/forum-2018-0133
Keywords for this article
CAT(0); dimension; rank rigidity

Articles in the same Issue

  1. Frontmatter
  2. Selberg integral over local fields
  3. On conjugation orbits of semisimple pairs in rank one
  4. On a topology property for the moduli space of Kapustin–Witten equations
  5. Differentiability of the evolution map and Mackey continuity
  6. A prime geodesic theorem for SL3(ℤ)
  7. Comparison and continuity of Wick-type star products on certain coadjoint orbits
  8. A construction of residues of Eisenstein series and related square-integrable classes in the cohomology of arithmetic groups of low k-rank
  9. Linear invariance of intersections on unitary Rapoport–Zink spaces
  10. Maximal subalgebras of finite-dimensional algebras
  11. A universal enveloping algebra for cocommutative rack bialgebras
  12. A rank rigidity result for CAT(0) spaces with one-dimensional Tits boundaries
  13. Projective objects in the category of pointwise finite dimensional representations of an interval finite quiver
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