The Dehn Function of Stallings’ Group

Abstract

We prove that the Dehn function of a group of Stallings that is finitely presented but not of type F 3 is quadratic.

Asymptotic invariants, complexity of groups and related problems

“…The evolution of a color 21 Note that Stallings' group was the first example of a finitely presented group which is not of type F P 3 . Thus by [71] and a previous results of Riley [203] and Papasoglu [196] this group is the first example of a finitely presented group with simply connected but not 2-connected asymptotic cones. 22 The R. Thompson group F consists of all piecewise-linear increasing bijective functions [0, 1] → [0, 1] whose derivatives have finitely many dyadic break points and the slopes of linear pieces are powers of 2.…”

“…The authors in [71] present an algorithm of creating a van Kampen diagram with quadratic area and a given boundary label.…”

Dehn functions of coabelian subgroups of direct products of groups

“…We say that a group G is of finiteness type F k if it admits a K(G, 1) which is a CW-complex with finitely many cells of dimension ≤ k. Historically, the first examples of groups of type F k−1 , not F k for all k ≥ 3, were the Stallings-Bieri groups SB k which arise as kernels SB k = ker(F 2 × ⋅ ⋅ ⋅ × F 2 → Z) of homomorphisms from products of k free groups onto Z [27,4]. Determining their precise Dehn functions turned out to be a challenging problem [19,2,7,17] that was only resolved in full generality very recently by Carter and Forester [12].…”

The Dehn functions of Stallings–Bieri groups

“…The first of the groups SB n for which the Dehn function is defined is SB 3 , also known as Stallings' group [18]. Dison, Elder, Riley, and Young [13] proved that SB 3 has quadratic Dehn function. Their method makes use of a particular presentation for SB 3 given in [3,15], and is essentially algebraic in nature.…”