Isoperimetric Inequalities for Homogeneous Nilpotent Groups

The large-scale geometry of locally compact solvable groups

“…It is worth mentioning that this filling inequality has a nice simple proof for connected nilpotent homogeneous Lie groups (outlined in [45, 5A' 5 ] and proved in details in [72]). Let us briefly summarize the argument.…”

Asymptotic invariants, complexity of groups and related problems

“…One can deduce the bound c + 1 + ε for every ε > 0 by the following argument. Pittet [199], proved that a lattice in a simply connected graded nilpotent Lie group of class c admits a polynomial coarse area function A δ of degree c + 1. Hence by the result of Pansu [194], every asymptotic cone of any nilpotent group of class c has isoperimetric function n c+1 .…”

“…The isoperimetric inequality n c+1 is the best possible bound for some nilpotent groups of class c. For example if G is a free nilpotent group of class c with at least two generators, then its Dehn function is polynomial of degree c + 1 (see [15] or [87] for the lower bound and [199] for the upper bound, see also [38] for other examples of nilpotent groups with maximal possible Dehn functions). Moreover, the following general theorem is proved in [15,99].…”

The Dehn function of Baumslag’s metabelian group

“…Much is already known about the isoperimetry of solvable groups. The Dehn functions of finitely generated nilpotent groups admit upper bounds of n c+1 where c is the nilpotency class [16,19], and yet for all c there are class c examples with Dehn function n c+1 [6,15,22] and others with Dehn function n 2 [24]. There are also nilpotent examples with Dehn function n 2 log n [23].…”