Metric properties of the group of area preserving diffeomorphisms. (English) Zbl 0977.57039
Let \(\mathcal{D}_2\) be the group of smooth area preserving diffeomorphisms of the 2-disk which are the identity near the boundary of the disk. A tangent vector to \(\mathcal{D}_2\) at a point \(\phi\) is a divergence free vector field \(X_\phi\). The \(L^2\)-norm on the Lie algebra of \(\mathcal{D}_2\) yields a right invariant metric on \(\mathcal{D}_2\). With this metric the diameter of \(\mathcal{D}_2\) is infinite [A. Shnirel’man, Math. USSR, Sb. 56, No. 1, 79-105 (1987); translation from Mat. Sb., Nov.Ser. 128(170), No. 1, 82-109 (1985; Zbl 0725.58005)]. Finitely generated groups are another standard class of groups equipped with a right invariant metric. Let \(\mathcal{G}\) be such a group, and \(\mathcal{S}\) a system of generators of \(\mathcal{G}\). The length \(l_S(g)\) of an element \(g\) in \(\mathcal{G}\) is the minimal number of generators and their inverses needed to write \(g\). It defines a right invariant metric on \(\mathcal{G}\): \(d_S(g_1,g_2)=l_S(g_1g_2^{-1})\). Let \((\mathcal{G}_1,d_1)\) and \((\mathcal{G}_2,d_2)\) be two metric groups; a morphism \(m:\mathcal{G}_1\rightarrow \mathcal{G}_2\) is a quasi-expanding embedding if: (i) \(m\) is injective; (ii) there exists a pair of positive constants \(k,k'>0\) such that for any pair \(g_1,g_1'\) in \(\mathcal{G}\): \(d_2(m(g_1),m(g_1'))\geq kd_1(g_1,g_1')-k'\). An isomorphism \(m\) between two metric groups is a quasi-isometry if \(m\) and its inverse are quasi-expanding embeddings.
Theorem. Any finitely generated free group and finitely generated abelian free group is quasi-isometrically embedding in \(\mathcal{D}_2\).
The keys for this theorem rely on the following two facts: (1) any finitely generated free group and finitely generated abelian free group can be mapped by a quasi-isometrically embedding in a braid group; (2) the braid subgroups obtained this way can be in turn mapped by a quasi-isometrical embedding in \(\mathcal{D}_2\). Many interesting results concerning these fields can be found in [A. M. Lukatskij, Sel. Math. Sov. 1, No. 2, 185-195 (1981; Zbl 0502.58009); Sov. Math., Dokl. 16, 70-74 (1975); translation from Dokl. Akad. Nauk SSSR 220, No. 2, 285-288 (1975; Zbl 0321.58013)].
Theorem. Any finitely generated free group and finitely generated abelian free group is quasi-isometrically embedding in \(\mathcal{D}_2\).
The keys for this theorem rely on the following two facts: (1) any finitely generated free group and finitely generated abelian free group can be mapped by a quasi-isometrically embedding in a braid group; (2) the braid subgroups obtained this way can be in turn mapped by a quasi-isometrical embedding in \(\mathcal{D}_2\). Many interesting results concerning these fields can be found in [A. M. Lukatskij, Sel. Math. Sov. 1, No. 2, 185-195 (1981; Zbl 0502.58009); Sov. Math., Dokl. 16, 70-74 (1975); translation from Dokl. Akad. Nauk SSSR 220, No. 2, 285-288 (1975; Zbl 0321.58013)].
MSC:
| 57S05 | Topological properties of groups of homeomorphisms or diffeomorphisms |
| 58D05 | Groups of diffeomorphisms and homeomorphisms as manifolds |
| 58B25 | Group structures and generalizations on infinite-dimensional manifolds |
| 76A02 | Foundations of fluid mechanics |
| 58B05 | Homotopy and topological questions for infinite-dimensional manifolds |
| 20F36 | Braid groups; Artin groups |
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