A metrizable topology on the contracting boundary of a group

Cashen, Christopher; Mackay, John

doi:10.1090/tran/7544

A metrizable topology on the contracting boundary of a group
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by Christopher H. Cashen and John M. Mackay PDF

Trans. Amer. Math. Soc. 372 (2019), 1555-1600 Request permission

Abstract:

The ‘contracting boundary’ of a proper geodesic metric space consists of equivalence classes of geodesic rays that behave like rays in a hyperbolic space. We introduce a geometrically relevant, quasi-isometry invariant topology on the contracting boundary. When the space is the Cayley graph of a finitely generated group we show that our new topology is metrizable.

References

Carolyn Abbott, Jason Behrstock, and Matthew Gentry Durham, Largest acylindrical actions and stability in hierarchically hyperbolic groups, preprint, arXiv:1705.06219, (2017).
Yael Algom-Kfir, Strongly contracting geodesics in outer space, Geom. Topol. 15 (2011), no. 4, 2181–2233. MR 2862155, DOI 10.2140/gt.2011.15.2181
Tarik Aougab, Matthew Gentry Durham, and Samuel J. Taylor, Pulling back stability with applications to $\textrm {Out}(F_n)$ and relatively hyperbolic groups, J. Lond. Math. Soc. (2) 96 (2017), no. 3, 565–583. MR 3742433, DOI 10.1112/jlms.12071
Goulnara N. Arzhantseva, Christopher H. Cashen, Dominik Gruber, and David Hume, Contracting geodesics in infinitely presented graphical small cancellation groups, preprint, arXiv:1602.03767v2, (2016).
Goulnara N. Arzhantseva, Christopher H. Cashen, Dominik Gruber, and David Hume, Characterizations of Morse quasi-geodesics via superlinear divergence and sublinear contraction, Doc. Math. 22 (2017), 1193–1224. MR 3690269
Jason Behrstock, A counterexample to questions about boundaries, stability, and commensurability, preprint, arXiv:1705.03984v2, (2017).
Jason A. Behrstock, Asymptotic geometry of the mapping class group and Teichmüller space, Geom. Topol. 10 (2006), 1523–1578. MR 2255505, DOI 10.2140/gt.2006.10.1523
Mladen Bestvina, Ken Bromberg, and Koji Fujiwara, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 1–64. MR 3415065, DOI 10.1007/s10240-014-0067-4
Mladen Bestvina and Koji Fujiwara, A characterization of higher rank symmetric spaces via bounded cohomology, Geom. Funct. Anal. 19 (2009), no. 1, 11–40. MR 2507218, DOI 10.1007/s00039-009-0717-8
B. H. Bowditch, Relatively hyperbolic groups, Internat. J. Algebra Comput. 22 (2012), no. 3, 1250016, 66. MR 2922380, DOI 10.1142/S0218196712500166
Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR 1744486, DOI 10.1007/978-3-662-12494-9
Christopher H. Cashen, Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology, Anal. Geom. Metr. Spaces 4 (2016), no. 1, 278–281. MR 3550299, DOI 10.1515/agms-2016-0011
Ruth Charney and Devin Murray, A rank-one CAT(0) group is determined by its Morse boundary, preprint, arXiv:1707.07028v1, (2017).
Ruth Charney and Harold Sultan, Contracting boundaries of $\rm CAT(0)$ spaces, J. Topol. 8 (2015), no. 1, 93–117. MR 3339446
Matthew Cordes, Morse boundaries of proper geodesic metric spaces, Groups Geom. Dyn. 11 (2017), no. 4, 1281–1306. MR 3737283, DOI 10.4171/GGD/429
Matthew Cordes, A survey on Morse boundaries & stability, preprint, arXiv:1704.07598v1, (2017).
Matthew Cordes and Matthew Gentry Durham, Boundary convex cocompactness and stability of subgroups of finitely generated groups, preprint, arXiv:1607.08899v1, (2016).
Matthew Cordes and David Hume, Stability and the Morse boundary, J. Lond. Math. Soc. (2) 95 (2017), no. 3, 963–988. MR 3664526, DOI 10.1112/jlms.12042
Christopher B. Croke and Bruce Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000), no. 3, 549–556. MR 1746908, DOI 10.1016/S0040-9383(99)00016-6
F. Dahmani, V. Guirardel, and D. Osin, Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245 (2017), no. 1156, v+152. MR 3589159, DOI 10.1090/memo/1156
Cornelia Druţu, Relatively hyperbolic groups: geometry and quasi-isometric invariance, Comment. Math. Helv. 84 (2009), no. 3, 503–546. MR 2507252, DOI 10.4171/CMH/171
Cornelia Druţu, Shahar Mozes, and Mark Sapir, Divergence in lattices in semisimple Lie groups and graphs of groups, Trans. Amer. Math. Soc. 362 (2010), no. 5, 2451–2505. MR 2584607, DOI 10.1090/S0002-9947-09-04882-X
Cornelia Druţu and Mark Sapir, Tree-graded spaces and asymptotic cones of groups, Topology 44 (2005), no. 5, 959–1058. With an appendix by Denis Osin and Mark Sapir. MR 2153979, DOI 10.1016/j.top.2005.03.003
Moon Duchin and Kasra Rafi, Divergence of geodesics in Teichmüller space and the mapping class group, Geom. Funct. Anal. 19 (2009), no. 3, 722–742. MR 2563768, DOI 10.1007/s00039-009-0017-3
Matthew Gentry Durham and Samuel J. Taylor, Convex cocompactness and stability in mapping class groups, Algebr. Geom. Topol. 15 (2015), no. 5, 2839–2859. MR 3426695, DOI 10.2140/agt.2015.15.2839
B. Farb, Relatively hyperbolic groups, Geom. Funct. Anal. 8 (1998), no. 5, 810–840. MR 1650094, DOI 10.1007/s000390050075
Daniel Groves and Jason Fox Manning, Dehn filling in relatively hyperbolic groups, Israel J. Math. 168 (2008), 317–429. MR 2448064, DOI 10.1007/s11856-008-1070-6
Ursula Hamenstädt, Rank-one isometries of proper $\textrm {CAT}(0)$-spaces, Discrete groups and geometric structures, Contemp. Math., vol. 501, Amer. Math. Soc., Providence, RI, 2009, pp. 43–59. MR 2581914, DOI 10.1090/conm/501/09839
G. Christopher Hruska, Relative hyperbolicity and relative quasiconvexity for countable groups, Algebr. Geom. Topol. 10 (2010), no. 3, 1807–1856. MR 2684983, DOI 10.2140/agt.2010.10.1807
Jason Fox Manning, The Bowditch boundary of $(G,\mathcal {H})$ when $G$ is hyperbolic, preprint, arXiv:1504.03630v2, (2015).
Howard A. Masur and Yair N. Minsky, Geometry of the complex of curves. I. Hyperbolicity, Invent. Math. 138 (1999), no. 1, 103–149. MR 1714338, DOI 10.1007/s002220050343
Yair N. Minsky, Quasi-projections in Teichmüller space, J. Reine Angew. Math. 473 (1996), 121–136. MR 1390685, DOI 10.1515/crll.1995.473.121
Devin Murray, Topology and dynamics of the contracting boundary of cocompact CAT(0) spaces, preprint, arXiv:1509.09314v2, (2015).
Alexander Yu. Ol′shanskii, Denis V. Osin, and Mark V. Sapir, Lacunary hyperbolic groups, Geom. Topol. 13 (2009), no. 4, 2051–2140. With an appendix by Michael Kapovich and Bruce Kleiner. MR 2507115, DOI 10.2140/gt.2009.13.2051
Denis V. Osin, Relatively hyperbolic groups: intrinsic geometry, algebraic properties, and algorithmic problems, Mem. Amer. Math. Soc. 179 (2006), no. 843, vi+100. MR 2182268, DOI 10.1090/memo/0843
D. Osin, Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368 (2016), no. 2, 851–888. MR 3430352, DOI 10.1090/tran/6343
P. Papasoglu, Strongly geodesically automatic groups are hyperbolic, Invent. Math. 121 (1995), no. 2, 323–334. MR 1346209, DOI 10.1007/BF01884301
Alessandro Sisto, Projections and relative hyperbolicity, Enseign. Math. (2) 59 (2013), no. 1-2, 165–181. MR 3113603, DOI 10.4171/LEM/59-1-6
Alessandro Sisto, Quasi-convexity of hyperbolically embedded subgroups, Math. Z. 283 (2016), no. 3-4, 649–658. MR 3519976, DOI 10.1007/s00209-016-1615-z
Harold Sultan, Hyperbolic quasi-geodesics in CAT(0) spaces, Geom. Dedicata 169 (2014), 209–224. MR 3175245, DOI 10.1007/s10711-013-9851-4
Hung Cong Tran, Divergence spectra and Morse boundaries of relatively hyperbolic groups, Geom. Dedicata 194 (2018), 99–129. MR 3798063, DOI 10.1007/s10711-017-0268-3