Decomposition theorems for asymptotic property C and property A. (English) Zbl 1416.54015
The authors define a coarse geometric property of metric spaces they call “finite APC-decomposition complexity”. This property interpolates between two properties “finite decomposition complexity” and “asymptotic property C” that are of great interest in coarse geometry. A space satisfying either of these two properties also has finite APC-decomposition complexity. On the other hand, the authors show that finite APC-decomposition complexity implies another well-known “property A”. The main theorem is about closure properties of the class of spaces with finite APC-decomposition complexity. For example, it is closed under direct products of spaces among other constructions. These results are of particular interest when applied to finitely generated groups with word metrics. In this situation there are also closure properties under some group-theoretic constructions such as amalgamated products and extensions.
Reviewer: Boris Goldfarb (Albany)
References:
| [1] | Beckhardt, S., Extension Properties of Asymptotic Property C and Finite Decomposition Complexity (2017), State University of New York at Albany, PhD thesis |
| [2] | Beckhardt, S.; Goldfarb, B., Extension properties of asymptotic property C and finite decomposition complexity, Topol. Appl., 239, 181-190 (2018) · Zbl 1400.20039 |
| [3] | Bell, G.; Dranishnikov, A., On asymptotic dimension of groups, Algebraic Geom. Topol., 1, 57-71 (2001) · Zbl 1008.20039 |
| [4] | Bell, G.; Moran, D., On constructions preserving the asymptotic topology of metric spaces, North Carolina J. Math. Stat., 1, 46-57 (2015) |
| [5] | Bell, G.; Moran, D.; Nagórko, A., Coarse property C and decomposition complexity, Topol. Appl., 227, 30-50 (2017) · Zbl 1372.51006 |
| [6] | Bell, G. C.; Nagórko, A., On the stability of asymptotic property C for products and some group extensions, Algebraic Geom. Topol., 18, 1, 221-245 (2018) · Zbl 1394.54018 |
| [7] | Bridson, M. R.; Haefliger, A., Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319 (1999), Springer-Verlag: Springer-Verlag Berlin · Zbl 0988.53001 |
| [8] | T. Davila, On asymptotic property C. ArXiv e-prints, Nov. 2016.; T. Davila, On asymptotic property C. ArXiv e-prints, Nov. 2016. |
| [9] | Dranishnikov, A.; Smith, J., Asymptotic dimension of discrete groups, Fundam. Math., 189, 1, 27-34 (2006) · Zbl 1100.20034 |
| [10] | Dranishnikov, A.; Zarichnyi, M., Asymptotic dimension, decomposition complexity, and Haver’s property C, Topol. Appl., 169, 99-107 (2014) · Zbl 1297.54064 |
| [11] | Dranishnikov, A.; Zarichnyi, M., Remarks on straight finite decomposition complexity, Topol. Appl., 227, 102-110 (2017) · Zbl 1372.54026 |
| [12] | Dranishnikov, A. N., Asymptotic topology, Usp. Mat. Nauk, 55, 6(336), 71-116 (2000) · Zbl 1028.54032 |
| [13] | J. Dydak, Decomposition complexity with respect to coarse properties. ArXiv e-prints, Nov. 2016.; J. Dydak, Decomposition complexity with respect to coarse properties. ArXiv e-prints, Nov. 2016. |
| [14] | Gromov, M., Asymptotic invariants of infinite groups, (Geometric Group Theory, vol. 2. Geometric Group Theory, vol. 2, Sussex, 1991. Geometric Group Theory, vol. 2. Geometric Group Theory, vol. 2, Sussex, 1991, London Math. Soc. Lecture Note Ser., vol. 182 (1993), Cambridge Univ. Press: Cambridge Univ. Press Cambridge), 1-295 |
| [15] | Guentner, E., Permanence in coarse geometry, (Recent Progress in General Topology. III (2014), Atlantis Press: Atlantis Press Paris), 507-533 · Zbl 1300.54003 |
| [16] | Guentner, E.; Tessera, R.; Yu, G., A notion of geometric complexity and its application to topological rigidity, Invent. Math., 189, 2, 315-357 (2012) · Zbl 1257.57028 |
| [17] | Guentner, E.; Tessera, R.; Yu, G., Discrete groups with finite decomposition complexity, Groups Geom. Dyn., 7, 2, 377-402 (2013) · Zbl 1272.52041 |
| [18] | D. Kasprowski, A. Nicas, D. Rosenthal, Regular finite decomposition complexity. ArXiv e-prints, Aug. 2016.; D. Kasprowski, A. Nicas, D. Rosenthal, Regular finite decomposition complexity. ArXiv e-prints, Aug. 2016. |
| [19] | Serre, J.-P., Trees, Springer Monographs in Mathematics (2003), Springer-Verlag: Springer-Verlag Berlin, Translated from the French original by John Stillwell, Corrected 2nd printing of the 1980 English translation · Zbl 1013.20001 |
| [20] | Yamauchi, T., Asymptotic property C of the countable direct sum of the integers, Topol. Appl., 184, 50-53 (2015) · Zbl 1311.54021 |
| [21] | Yu, G., The coarse Baum-Connes conjecture for spaces which admit a uniform embedding into Hilbert space, Invent. Math., 139, 1, 201-240 (2000) · Zbl 0956.19004 |
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