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A topologically rigid set of quotients of the Davis complex

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Abstract

A class of topological spaces is topologically rigid if any two spaces with the same fundamental group are also homeomorphic. Topological rigidity, in addition to its intrinsic interest, has been useful for solving abstract commensurability questions. In this paper, we explore the topological rigidity of quotients of the Davis complex of certain right angled Coxeter groups by providing conditions on the defining graphs that obstruct topological rigidity. Furthermore, we explore why topological rigidity is hard to achieve for quotients of the Davis complex. Nonetheless, we conclude by introducing infinitely many infinite topologically rigid subclasses.

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References

  1. Crisp, John, Paoluzzi, Luisa: Commensurability classification of a family of right-angled coxeter groups. Proc. Amer. Math. Soc. 136(7), 2343–2349 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  2. Dani, Pallavi, Stark, Emily, Thomas, Anne: Commensurability for certain right-angled Coxeter groups and geometric amalgams of free groups. Groups Geomet. Dyn. 12(4), 1273–1341 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  3. Davis, Michael W.: The geometry and Topology of Coxeter Groups London Mathematical Society Monographs Series. Princeton University Press, Princeton, NJ (2008)

    Google Scholar 

  4. Lafont, Jean-Francois.: Rigidity result for certain 3-dimensional singular spaces and their fundamental groups. Geom. Dedicata 109, 197–219 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  5. Lafont, Jean-Francois.: Diagram rigidity for geometric amalgamations of free groups. J. Pure Appl. Algebra 209(3), 771–780 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Lafont, Jean-Francois.: Rigidity of hyperbolic P-manifolds: a survey. Geom. Dedicata 124, 143–152 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  7. Stark, Emily: Topological rigidity fails for quotients of the Davis orbicomplex. Proc. Amer. Math. Soc. 146(12), 5357–5366 (2018)

    Article  MathSciNet  MATH  Google Scholar 

  8. Arthur T. White. Imbedding Problems in Graph Theory. 6 in Graphs of Groups on Surfaces: Interactions and Models (Ed. A. T. White). Elsevier: Amsterdam, Netherlands. pp 49-72 (2001)

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Acknowledgements

The author would like to thank Tullia Dymarz for discussing this work in depth and editing many preliminary drafts. The author would also like to thank the anonymous referee for very helpful comments on a draft of the paper and especially for pointing out edge cases in the proofs of Theorems 1 and 2. Additionally, the author thanks Emily Stark for helpful discussions.

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  1. Department of Mathematics, University of Wisconsin, Madison, 53706, USA

    Yandi Wu

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  1. Yandi Wu
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Correspondence to Yandi Wu.

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Wu, Y. A topologically rigid set of quotients of the Davis complex. Geom Dedicata 217, 82 (2023). https://doi.org/10.1007/s10711-023-00819-6

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