Homological Dehn functions of groups of type $FP_2$

Abstract

We prove foundational results for homological Dehn functions of groups of type F P 2 such as superadditivity and the invariance under quasi-isometry. We then study the homological Dehn functions for Leary's groups G L (S) providing methods to obtain uncountably many groups with a given homological Dehn function. This allows us to show that there exist groups of type F P 2 with quartic homological Dehn function and unsolvable word problem.

Dehn functions of coabelian subgroups of direct products of groups

“…Our proofs provide a constructive way of filling a loop with a disk. One can apply a similar construction for the homological finite presentations from [5] to bound the homological Dehn function when each 𝐺 𝑖 is of type 𝐹𝑃 2 . More precisely, if in Theorem 3.2 or Theorem 4.2 we assume that each 𝐺 𝑖 is of type 𝐹𝑃 2 (rather than finitely presented), then 𝐾 is of type 𝐹𝑃 2 and 𝑓(𝑛) ≼ 𝛿 𝐾 (𝑛) ≼ 𝑓(𝑛) ⋅ 𝑙𝑜g(𝑛), where 𝑓 is the homological Dehn function of ∏ 𝑖 𝐺 𝑖 .…”

“…For the upper bound, one uses that homological Dehn functions are always superadditive [5,Proposition 2.20].…”

Superexponential Dehn functions inside CAT(0) groups

“…The proof of the next proposition uses the machinery of singular disk diagrams and combinatorial 2-complexes. The reader may wish to refer to Section 2 of [BKS20] for background on combinatorial complexes and singular disk fillings of combinatorial loops; specifically, Definitions 2.4, 2.9, 2.10, and 2.11. We need to bound the area of an edgepath loop in X as a function of its length.…”

On the virtual and residual properties of a generalization of Bestvina-Brady groups

“…Bestvina-Brady groups are a family of infinite discrete groups that were constructed in the 1990s to answer a long standing open question in homological group theory [2]; the existence of non-finitely presented groups of type F P. In [10], one of us generalized the Bestvina-Brady construction, producing an uncountable family of groups of type F P. Further results concerning these groups can be found in [3,8]. Our aim is to study some of the other, non-homological, properties of these groups.…”