Dehn functions of subgroups of products of free groups. Part I: Uniform upper bounds

Abstract

Subgroups of direct products of finitely many finitely generated free groups form a natural class that plays an important role in geometric group theory. Its members include fundamental examples, such as the Stallings-Bieri groups. This raises the problem of understanding their geometric invariants. We prove that finitely presented subgroups of direct products of three free groups, as well as subgroups of finiteness type ${\mathcal{F}}_{n-1}$ in a direct product of $n$ free groups, have Dehn function bounded above by $N^9$. This gives a positive answer to a question of Dison within these important subclasses and provides new insights in the context of Bridson's conjecture stating that finitely presented subgroups of direct products of free groups have polynomially bounded Dehn function. To prove our results we generalise techniques for "pushing fillings" into normal subgroups.

v2: We split the paper into two parts. This is part I, containing the results on uniform upper bounds (Theorems A, C and D of v1). Part II will contain the precise computations of Dehn functions (Theorem B and E of v1). 46 pages, 1 figure