Homological and homotopical Dehn functions are different

Abstract

The homological and homotopical Dehn functions are different ways of measuring the difficulty of filling a closed curve inside a group or a space. The homological Dehn function measures fillings of cycles by chains, whereas the homotopical Dehn function measures fillings of curves by disks. Because the two definitions involve different sorts of boundaries and fillings, there is no a priori relationship between the two functions; however, before this work, there were no known examples of finitely presented groups for which the two functions differ. This paper gives such examples, constructed by amalgamating a free-by-cyclic group with several Bestvina-Brady groups.

Snowflake geometry in CAT (0) groups

“…As in Claim 1, if w i satisfies (3) or (4) then Claim 2 is trivially true. The remaining case is when w i satisfies (2). Now the claim is an instance of the induction hypothesis: since v i is reduced we have v i = w i Aν 0 K 3 w i α .…”

“…This is simply the base case of the induction. • Case IB: either there is a subword w i of type (1), or there are two or more subwords of type (2). • Case II: exactly one subword w i is of type (2) and all others are of types (3) or (4).…”

“…) C(3|α|)2 and Area(Δ β ) C(3|β|)2 .Two more loops remain to be filled. Letŵ(a 0 , b 0 ) be the word labeling [p 1 ,p 2 ] and [o 1 , o 2 ].…”

Finiteness of homological filling functions

“…Under the assumptions, the group G is finitely presented relative to P if there is a finite subset R ⊂ F (S) * P such that the kernel of the map F (S) * P → G is the smallest normal subgroup containing R. In this case, we say that (1) S, P |R is a finite relative presentation of G with respect to P . It is an exercise to show that if G is finitely presented and P is finitely generated, then G is finitely presented relative to P .…”

Subgroups of relatively hyperbolic groups of Bredon cohomological dimension 2

“…The homological Dehn function of a space is a generalized isoperimetric function describing the minimal volume required to fill cellular 1-cycles with cellular 2-chains, see Section 4 for definitions. The homological Dehn function FV G : N → N of a finitely presented group G is the homological Dehn function of the universal cover of a K(G, 1) with finite 2-skeleton; the growth rate of this function is an invariant of the group [1,13,15].…”