On quadratic Dehn functions

摘要

We confirm with new examples that "Solvable groups of high R-rank are expected to satisfy a polynomial isoperimetric inequality" ([Gro93] 5A9). To that end we study invariant quasi-geodesic foliations in simply connected solvable Lie groups, endowed with left-invariant Riemannian metrics, whose leaves are isometric to closed subgroups. We establish a decomposition theorem which implies upper bounds on the Dehn (or filling) function (of loops by disks) of the solvable group in terms of the Dehn functions of the leaves. We obtain examples of metabelian polycyclic groups with exponential growth and quadratic Dehn functions. We also deduce that the horospheres in SL(4, R)/SO(4, R) which bound an invariant core for SL(4, Z) and that the horospheres which bound an invariant core for Hilbert modular groups in (H-2)(n), n > 2, have quadratic filling functions. The main theorem also applies to some solvable Lie groups which are not quasi-isometric to horospheres in symmetric spaces.