Filling Inequalities for Nilpotent Groups

We give methods for bounding the higher-order filling functions of a homogeneous nilpotent group and apply them to a family of quadratically presented groups constructed by S. Chen[9]. We find sharp bounds on some higher-order filling invariants of these groups. In particular, we show that groups with arbitrarily large nilpotency class can have euclidean n-dimensional filling volume and give an infinite family of groups with arbitrarily large nilpotency class and quadratic Dehn functions. Dehn functions of groups measure the complexity of the word problem for a group by counting the number of relators necessary to reduce a word to the identity. This can be interpreted geometrically as a problem of filling closed curves in a corresponding space with discs. One particularly interesting case is homogeneous nilpotent groups; in this case, the geometry of the group is closely connected to a corresponding Carnot-Caratheodory space, and efficient fillings of curves correspond to discs satisfying certain tangency conditions. In this paper, we will use this connection to find higher-order filling functions for some nilpotent groups. Previously, these functions were known only for abelian groups[12] and the topdimensional filling functions of nilpotent groups[19, IV.7]. We will also find the smallest possible Dehn function for a torsion-free group of a given nilpotency class by exhibiting a family of groups of arbitrary nilpotency class but quadratic Dehn function. The largest possible Dehn function for a nilpotent group of a given class is addressed by Gersten’s conjecture that all c-step nilpotent group satisfy an isoperimetric inequality of degree c+1; this was proven by Gromov[15, 5.A5] using infinitesimally invertible operators and combinatorially by Gersten, Holt, and Riley[13]. On the other hand, Gromov proved in [14](see also [16]) that a group has a linear Dehn function if and only if it is hyperbolic and that any non-hyperbolic group has a Dehn function which grows at least quadratically. Thus, the Dehn function of any torsion-free non-abelian nilpotent group is at least quadratic. There are several examples of two-step groups achieving this bound; our main goal in this paper is to give examples of groups of arbitrary nilpotency class which achieve this bound. One example of a family of two-step nilpotent groups with quadratic Dehn function is the higher-dimensional Heisenberg groups(this result was stated by Thurston[11] and proved by Gromov[15], Allcock[1], and Sapir and Olshanskii[17]). In addition, Gromov showed that having quadratic Dehn function is in fact generic for two-step nilpotent Lie groups that satisfy an inequality involving their rank and that of their abelianization. There are few known examples of nilpotent groups of class 3 or higher with quadratic Dehn function. An example of such a class 3 group, Date: March 11, 2008.