On quadratic Dehn functions

Abstract

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.

Nilpotent groups without exactly polynomial Dehn function

“…In [24] it is also claimed that the octonionic Heisenberg group has cubic Dehn function. As was pointed out in [17] there is a sign error in the proof of the octonionic case, so that the best previously known lower bound is quadratic. From Theorem 5.1 we obtain a super-quadratic lower bound.…”

Metabelian Groups With Quadratic Dehn Function and Baumslag–solitar Groups

“…Quadratic Dehn function groups enjoy some special properties not shared by all polynomial Dehn function groups, for instance they have all asymptotic cones simply connected [20]. After Gromov [13], many solvable groups were proved to have quadratic Dehn function (see for instance [2,12,21]). …”

The large-scale geometry of locally compact solvable groups

“…However, estimates on horospherical groups are still useful. Young [105] uses quadratic filling inequalities proved by Leuzinger and Pittet [63] for certain solvable Lie groups in his recent groundbreaking demonstration that SL(n, Z) has quadratic Dehn function for n ≥ 5. This fact has been conjectured by Thurston [27] for n ≥ 4.…”