×

Thompson’s group \(F\) and the linear group \(\mathrm{GL}_\infty(\mathbb Z)\). (English) Zbl 1262.20049

Summary: The authors study the finite decomposition complexity of metric spaces of \(H\), equipped with different metrics, where \(H\) is a subgroup of the linear group \(\mathrm{GL}_\infty(\mathbb Z)\). It is proved that there is an injective Lipschitz map \(\varphi\colon(F,d_S)\to(H,d)\), where \(F\) is Thompson’s group, \(d_S\) the word-metric of \(F\) with respect to the finite generating set \(S\) and \(d\) a metric of \(H\). But it is not a proper map. Meanwhile, it is proved that \(\varphi\colon(F,d_S) \to(H,d_1)\) is not a Lipschitz map, where \(d_1\) is another metric of \(H\).

MSC:

20F69 Asymptotic properties of groups
20H25 Other matrix groups over rings
20F65 Geometric group theory
20F05 Generators, relations, and presentations of groups
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Gromov, M., Asymptotic invariants of infinite groups, Geometric Group Theory, London Math. Soc., Lecture Notes, Vol. 2, Ser. 182, Cambridge Univ. Press, Cambridge, 1993. · Zbl 0841.20039
[2] Tessera, R., Guentner, E. and Yu, G., A notion of geometric complexity and its application to topological rigidity, 2010. arXiv:1008.0884v1 · Zbl 1257.57028
[3] Willett, R., Some Notes on Property A, Limits of Graphs in Group Theory and Computer Science, EPFL Press, Lausanne, 2009, 191–281. · Zbl 1201.19002
[4] Bell, G. and Dranishnikov, A., Asymptotic dimension in Będlewo, Topol. Proc., 38, 2011, 209–236. · Zbl 1261.20044
[5] Belk, J. M., Thompson’s Group F, PhD Thesis, Cornell University, Ithaca, New York, 2004. arXiv: math.GR/0708.3609v1
[6] Fordham, S. B., Minimal Length Elements of Thompson’s Group F, PhD Thesis, Brigham Young University, Provo, Utah, 1995. · Zbl 1039.20014
[7] Floyd, W. J., Cannon, J. W. and Parry, W. R., Introductory notes on Richard Thompson’s groups, L’Enseign. Math. (2), 42(3–4), 1996, 215–256. · Zbl 0880.20027
[8] Cleary, S. and Taback, J., Combinatorial properties of Thompson’s group F, Trans. Amer. Math. Soc., 356(7), 2004, 2825–2849. · Zbl 1065.20052 · doi:10.1090/S0002-9947-03-03375-0
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.