Thompson’s group F and the linear group GL _∞(ℤ)

Abstract

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 GL_∞(ℤ). It is proved that there is an injective Lipschitz map φ: (F, d _S) → (H, d), where F is the Thompson’s group, dS 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 φ: (F, d _S) → (H, d ₁) is not a Lipschitz map, where d ₁ is another metric of H.

Keywords

Finite decomposition complexity / Thompson’s group F / Word-metric / Lipschitz map / Reduced tree diagram

References

[1] Gromov M.. Asymptotic invariants of infinite groups, 1993, Cambridge: Cambridge Univ. Press

[2] Tessera, R., Guentner, E. and Yu, G., A notion of geometric complexity and its application to topological rigidity, 2010. arXiv:1008.0884v1

[3] Willett R.. Some Notes on Property A, Limits of Graphs in Group Theory and Computer Science, 2009, Lausanne: EPFL Press 191-281

[4] Bell G., Dranishnikov A.. Asymptotic dimension in Będlewo. Topol. Proc., 2011, 38: 209-236

[5] Belk J. M.. Thompson’s Group F, 2004, Ithaca, New York: Cornell University

[6] Fordham S. B.. Minimal Length Elements of Thompson’s Group F, 1995, Provo, Utah: Brigham Young University

[7] Floyd W. J., Cannon J. W., Parry W. R.. Introductory notes on Richard Thompson’s groups. L’Enseign. Math. (2), 1996, 42(3–4): 215-256

[8] Cleary S., Taback J.. Combinatorial properties of Thompson’s group F. Trans. Amer. Math. Soc., 2004, 356(7): 2825-2849