Isoperimetric inequalities for nilpotent groups

Gersten, S. M.; Holt, Derek F.; Riley, Tim

doi:10.1007/s00039-003-0430-y

Cited by 22 publications

(30 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…This would make the result applicable iteratively, and we would therefore be able to constrain both the Dehn function and the gallery length function as one takes successive central extensions. As a corollary we would reproduce the result of [13][14][15] that finitely generated nilpotent groups of class c admit polynomial isoperimetric functions of degree c + 1.…”

Section: Conjecture 54supporting

confidence: 59%

“…If Conjecture 4.3 holds, then by Theorem 1.3 the linear bound on the gallery length would also recapture the linear bound on filling length of [13,20].…”

Section: Corollary 66 Assume Conjecture 43 Holds If P Is a Finitementioning

confidence: 99%

“…The filling length function is discussed extensively in [10] and has an important application in [13]. We introduced GL and DGL in [12], and DlogA, which will play an important role in § 7, is new.…”

Section: Filling Functions For Finite Presentationsmentioning

confidence: 99%

“…And FL P (w) for a word w that represents 1 in the group presented by P is the minimum of FL(D) over all van Kampen diagrams D for w. And then FL P (n) is the maximum of FL P (w) quantifying over all words w of length at most n that represent 1 in the group. It is possible to interpret FL P as the non-deterministic space-complexity of a naive approach to solving the word problem for P in which relators are applied exhaustively (see § 5 or [13] for more details).…”

Section: Conjecture 54mentioning

confidence: 99%

“…Finitely generated nilpotent groups are also known to have filling length functions that admit linear upper bounds [13][14][15]. It is an open question whether or not such groups are asynchronously combable.…”

Section: Estimates On Gallery Length and Filling Lengthmentioning

confidence: 99%

See 4 more Smart Citations

Some Duality Conjectures for Finite Graphs and Their Group Theoretic Consequences

Gersten¹,

Riley²

2005

Proceedings of the Edinburgh Mathematical Society

View full text Add to dashboard Cite

We pose some graph theoretic conjectures about duality and the diameter of maximal trees in planar graphs, and we give innovations in the following two topics in geometric group theory, where the conjectures have applications.Central extensions. We describe an electrostatic model concerning how van Kampen diagrams change when one takes a central extension of a group. Modulo the conjectures, this leads to a new proof that finitely generated class c nilpotent groups admit degree c + 1 polynomial isoperimetric functions.Filling functions. We collate and extend results about interrelationships between filling functions for finite presentations of groups. We use the electrostatic model in proving that the gallery length filling function, which measures the diameter of the duals of diagrams, is qualitatively the same as a filling function DlogA, concerning the sum of the diameter with the logarithm of the area of a diagram. We show that the conjectures imply that the space-complexity filling function filling length essentially equates to gallery length. We give linear upper bounds on these functions for a number of classes of groups including fundamental groups of compact geometrizable 3-manifolds, certain graphs of groups, and almost convex groups. Also we define restricted filling functions which concern diagrams with uniformly bounded vertex valence, and we show that, assuming the conjectures, they reduce to just two filling functions-the analogues of non-deterministic space and time.

show abstract

Section: Conjecture 54supporting

confidence: 59%

“…If Conjecture 4.3 holds, then by Theorem 1.3 the linear bound on the gallery length would also recapture the linear bound on filling length of [13,20].…”

Section: Corollary 66 Assume Conjecture 43 Holds If P Is a Finitementioning

confidence: 99%