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Super-exponential distortion of subgroups of \(\text{CAT}(-1)\) groups. (English) Zbl 1187.20055
From the introduction: The purpose of this note is to produce explicit examples of \(\text{CAT}(-1)\) groups containing free subgroups with arbitrary iterated exponential distortion, and distortion higher than any iterated exponential. The construction parallels that of M. Mitra [in J. Differ. Geom. 48, No. 1, 135-164 (1998; Zbl 0906.20023)] but our groups are the fundamental groups of locally \(\text{CAT}(-1)\) 2-complexes. The building blocks used in [loc. cit.] are hyperbolic \(F_3\rtimes F_3\) groups, which are not known to be \(\text{CAT}(0)\). Our building blocks are graphs of groups where the vertex and edge groups are all free groups of equal rank and the underlying graph is a bouquet of a finite number of circles. We use the combinatorial and geometric techniques from D. T. Wise’s version of the Rips construction [Proc. Am. Math. Soc. 126, No. 4, 957-964 (1998; Zbl 0913.20027)] to ensure that our building blocks glue together in a locally \(\text{CAT}(-1)\) fashion.

MSC:
20F65 Geometric group theory
20F67 Hyperbolic groups and nonpositively curved groups
57M07 Topological methods in group theory
20E06 Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations
57M20 Two-dimensional complexes (manifolds) (MSC2010)
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References:
[1] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften 319, Springer (1999) · Zbl 0988.53001
[2] M Mitra, Cannon-Thurston maps for trees of hyperbolic metric spaces, J. Differential Geom. 48 (1998) 135 · Zbl 0906.20023
[3] D T Wise, Incoherent negatively curved groups, Proc. Amer. Math. Soc. 126 (1998) 957 · Zbl 0913.20027 · doi:10.1090/S0002-9939-98-04146-X
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