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On quadratic Dehn functions

On quadratic Dehn functions 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 [InlineMediaObject not available: see fulltext.] have quadratic filling functions. The main theorem also applies to some solvable Lie groups which are not quasi-isometric to horospheres in symmetric spaces. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

On quadratic Dehn functions

Mathematische Zeitschrift , Volume 248 (4) – May 6, 2004

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References (23)

Publisher
Springer Journals
Copyright
Copyright © 2004 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
DOI
10.1007/s00209-004-0678-4
Publisher site
See Article on Publisher Site

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 [InlineMediaObject not available: see fulltext.] have quadratic filling functions. The main theorem also applies to some solvable Lie groups which are not quasi-isometric to horospheres in symmetric spaces.

Journal

Mathematische ZeitschriftSpringer Journals

Published: May 6, 2004

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