Quasi-isometries need not induce homeomorphisms of contracting boundaries with the Gromov product topology. (English) Zbl 1377.20030
The paper under review gives examples that quasi-isometries of \(\mathrm{CAT}(0)\) spaces do not, in general, induce homeomorphisms of the contracting boundary with subspace topology. The notion of a quasi-isometry invariant of a contracting boundary for a \(\mathrm{CAT}(0)\) space was introduced by R. Charney and H. Sultan [J. Topol. 8, No. 1, 93–117 (2015; Zbl 1367.20043)] and it enjoys many of the properties satisfied by boundaries of hyperbolic spaces.
The author’s example produces a quasi-isometry between \(\mathrm{CAT}(0)\) spaces but does not induces a continuous map between contracting boundaries. Because of this, the author proposes the following question:
“If \(\phi\) is a quasi-isometry between proper geodesic spaces \(X\) and \(Y\) that have a cocompact isometry groups and such that \(X\) and \(Y\) are \(\mathrm{CAT}(0)\), must \(\partial_C\phi : \partial_C^{G_p} X \rightarrow \partial_C^{G_p} Y\) be a homeomorphism?”
The author’s example produces a quasi-isometry between \(\mathrm{CAT}(0)\) spaces but does not induces a continuous map between contracting boundaries. Because of this, the author proposes the following question:
“If \(\phi\) is a quasi-isometry between proper geodesic spaces \(X\) and \(Y\) that have a cocompact isometry groups and such that \(X\) and \(Y\) are \(\mathrm{CAT}(0)\), must \(\partial_C\phi : \partial_C^{G_p} X \rightarrow \partial_C^{G_p} Y\) be a homeomorphism?”
Reviewer: Fabio Santos (Rio de Janeiro)
MSC:
20F65 | Geometric group theory |
20F67 | Hyperbolic groups and nonpositively curved groups |
20F55 | Reflection and Coxeter groups (group-theoretic aspects) |
53C22 | Geodesics in global differential geometry |
57M07 | Topological methods in group theory |
Keywords:
Gromov boundary; quasi-isometry; contracting boundary; \(\mathrm{CAT}(0)\) space; geodesic spacesCitations:
Zbl 1367.20043References:
[1] | [1] Goulnara N. Arzhantseva, Christopher H. Cashen, Dominik Gruber, and David Hume, Characterizations of Morse geodesics via superlinear divergence and sublinear contraction, preprint (2016), arXiv:1601.01897.; · Zbl 1483.20077 |
[2] | Martin R. Bridson and André Haefliger, Metric spaces of non-positive curvature, Grundlehren der mathematischen Wissenschaften, vol. 319, Springer, Berlin, 1999.; · Zbl 0988.53001 |
[3] | Stephen M. Buckley and Simon L. Kokkendorff, Comparing the Floyd and ideal boundaries of a metric space, Trans. Amer. Math. Soc. 361 (2009), no. 2, 715-734.; · Zbl 1182.54030 |
[4] | Ruth Charney and Harold Sultan, Contracting boundaries of CAT(0) spaces, J. Topol. 8 (2015), no. 1, 93-117.; · Zbl 1367.20043 |
[5] | Matthew Cordes, Morse boundaries of proper geodesic metric spaces, preprint (2015), arXiv:1502.04376.; · Zbl 1423.20043 |
[6] | Christopher B. Croke and Bruce Kleiner, Spaces with nonpositive curvature and their ideal boundaries, Topology 39 (2000), 549-556.; · Zbl 0959.53014 |
[7] | M. Gromov, Hyperbolic groups, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75- 263.; · Zbl 0634.20015 |
[8] | Harold Sultan, Hyperbolic quasi-geodesics in CAT(0) spaces, Geom. Dedicata 169 (2014), no. 1, 209-224.; · Zbl 1330.20065 |
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