A subgroup of a direct product of free groups whose Dehn function has a cubic lower bound

Abstract

Abstract. We establish a cubic lower bound on the Dehn function of a certain finitely presented subgroup of a direct product of three free groups.

Dehn functions of coabelian subgroups of direct products of groups

“…We recall that the bound on the corank in Theorem 5.1 is optimal: Dison proved that the kernel of the canonical homomorphism F 2 ×F 2 ×F 2 → Z 2 induced by the abelianization on factors satisfies a cubical lower bound on its Dehn function [16], while Theorem 5.1 shows that for r ≥ 4 the kernel of the canonical homomorphism F ×r 2 → Z 2 induced by the abelianization on factors has quadratic Dehn function.…”

“…Note that this result is optimal in r for the subfamily K r 2 (2), r ≥ 3, since K 3 2 (2) satisfies a cubic lower bound on its Dehn function [16]. Theorem 1.5 will follow from the computation of the precise Dehn function for a more general class of coabelian SPFs (see Theorem 5.1).…”

“…The number of factors here cannot be reduced. Indeed, the kernel of the homomorphism φ∶ F 2 × F 2 × F 2 → Z 2 defined by abelianization on the factors satisfies a cubic lower bound on its Dehn function [16].…”

Dehn functions of coabelian subgroups of direct products of groups

“…The number of factors here cannot be reduced. Indeed, the kernel of the homomorphism $$\phi \colon F_2\times F_2\times F_2\rightarrow {\mathbb {Z}}^2$$ defined by abelianisation on the factors satisfies a cubic lower bound on its Dehn function [18]. Remark Our proofs provide a constructive way of filling a loop with a disk.…”

“…Note that this result is optimal in $$r$$ for the subfamily $$K_2^r(2)$$, $$r\geqslant 3$$, since $$K_2^3(2)$$ satisfies a cubic lower bound on its Dehn function [18]. Theorem 1.5 will follow from the computation of the precise Dehn function for a more general class of coabelian SPFs (see Theorem 5.1).…”