Abstract
Behrstock and Druţu raised a question about the existence of Morse geodesics in \({{\mathrm{CAT}}}(0)\) spaces with divergence function strictly greater than \(r^n\) and strictly less than \(r^{n+1}\), where n is an integer \({>}1\). In this paper, we answer the question of Behrstock and Druţu by showing that for each real number \(s\ge 2\), there is a \({{\mathrm{CAT}}}(0)\) space X with a proper and cocompact action of some finitely generated group such that X contains a Morse bi-infinite geodesic with the divergence equivalent to \(r^s\).
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References
Behrstock, J., Charney, R.: Divergence and quasimorphisms of right-angled Artin groups. Math. Ann. 352(2), 339–356 (2012)
Behrstock, J., Druţu, C.: Divergence, thick groups, and short conjugators. Preprint. arXiv:1110.5005
Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Volume 319 of Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences]. Springer, Berlin (1999)
Charney, R., Sultan, H.: Contracting boundaries of \(\text{ CAT }(0)\) spaces. J. Topol. 8(1), 93–117 (2015)
Davis, M.W.: The Geometry and Topology of Coxeter Groups, Volume 32 of London Mathematical Society Monographs Series. Princeton University Press, Princeton (2008)
Druţu, C., Mozes, S., Sapir, M.: Divergence in lattices in semisimple Lie groups and graphs of groups. Trans. Am. Math. Soc. 362(5), 2451–2505 (2010)
Duchin, M., Rafi, K.: Divergence of geodesics in Teichmüller space and the mapping class group. Geom. Funct. Anal. 19(3), 722–742 (2009)
Dani, P., Thomas, A.: Divergence in right-angled Coxeter groups. Trans. Am. Math. Soc. 367(5), 3549–3577 (2015)
Gersten, S.M.: Quadratic divergence of geodesics in \(\text{ CAT }(0)\) spaces. Geom. Funct. Anal. 4(1), 37–51 (1994)
Lyndon, R.C., Schupp, P.E.: Combinatorial Group Theory. Springer, Berlin (1977). Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 89
Macura, N.: CAT(0) spaces with polynomial divergence of geodesics. Geom. Dedicata 163, 361–378 (2013)
Sisto, A.: On metric relative hyperbolicity. Preprint. arXiv:1210.8081
Tran, H.: Relative divergence of finitely generated groups. Algebr. Geom. Topol. 15(3), 1717–1769 (2015)
Acknowledgments
I would like to thank my advisor Prof. Christopher Hruska for very helpful comments and suggestions. I also thank the referee for the helpful advice and the suggestion that improves the main theorem in the article. I want to thank Pallavi Dani, Jason Behrstock, and Hoang Thanh Nguyen for their helpful correspondences.
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Tran, H.C. Divergence of Morse geodesics. Geom Dedicata 180, 385–397 (2016). https://doi.org/10.1007/s10711-015-0107-3
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DOI: https://doi.org/10.1007/s10711-015-0107-3