Nilpotent groups without exactly polynomial Dehn function

Abstract

We prove super-quadratic lower bounds for the growth of the filling area function of a certain class of nilpotent Lie groups. This class contains groups for which it is known that their Dehn function grows no faster than n 2 log n. We therefore obtain the existence of (finitely generated) nilpotent groups whose Dehn functions do not have exactly polynomial growth and we thus answer a well-known question about the possible growth rate of Dehn functions of nilpotent groups.

Asymptotic invariants, complexity of groups and related problems

“…Note that Theorem 2.28 does not give metric properties of asymptotic cones, only the topological property of being simply connected. In a simply connected asymptotic cone even of a nice nilpotent group, a Lipschitz loop does not necessarily bound a Lipschitz 2-disc or even an integral 2-current (it can be deduced from [6] for the Heisenberg group H 3 , for other examples see Wenger [235], and 3.2.C). But for groups with quadratic Dehn functions, the asymptotic cones are much nicer.…”

“…Then we were not able to prove that the central square of G 10 has quadratic Dehn function (this example and the question can be found in [237,Section 5]). [235] showed that in fact central squares of nilpotent groups of class 2 often do not admit quadratic isoperimetric inequality. In particular, he proved Theorem 3.36 (Wenger [235]) The Dehn function of the central square of G 10 is strictly greater than quadratic.…”

“…This definition was used to obtain remarkable results in [234,235]. Since it is more complicated than the other two definitions, we present only some ideas it is based on, referring the reader to [5,43].…”

“…In fact there are several kinds of definitions of area in that case. Below, we shall mention the definition of Bowditch [28, Sections 2.3, 5; 29] Gromov's definition from of coarse area function from [99,5.F] and a definition using metric currents [5,234,235]. In all these definitions (X, dist) is a geodesic metric space.…”

“…Still there are very many new results and ideas that appeared after the last of these surveys was published. We can mention, for example, the construction of finitely presented groups with polynomial-non-recursive and even quadratic-non-recursive Dehn functions [182], finding a nilpotent finitely generated group with Dehn function not of the form n α for any α [235], the proof that all groups SL n (Z), n ≥ 5, the R. Thompson group F, extensions of finitely generated free groups by cyclic groups, have quadratic Dehn functions [36,111,239], the description of all FFFL (space) functions of finitely presented groups and the algebraic characterization of groups with PSPACE word problem [183], solving the isomorphism problem for arbitrary hyperbolic groups [65] and many others. Several new methods are used in the proofs of these results and our goal in this survey is to give as gentle as possible an introduction to these results and methods.…”