Lower bounds on cubical dimensionof $C’(1/6)$ groups

Abstract

For each n we construct examples of finitely presented C ′ (1/6) small cancellation groups that do not act properly on any n-dimensional CAT(0) cube complex.

Groups with arbitrary cubical dimension gap

“…We finish this section with an observation to the examples constructed in [7]. Namely, the examples can be used to construct aspherical 4manifolds with an arbitrary dimension gap.…”

“…Proof. Let H n be the group constructed in [7] which does not act properly on any CAT(0) cube complex of dimension ă n. The group H n is a small cancellation group so the presentation 2-complex is a classifying space. We can embed this classifying space into R 4 [10] and take a neighbourhood to get a 4-manifold which is a classifying space.…”

“…We use this cubical splitting theorem to obtain groups with arbitrary gaps between the CAT(0) dimension and cubical dimension. Examples of this kind have previously been given by [7] using small cancellation theory. We build on the work of [3] to give examples with explicit complexes realising both the CAT(0) dimension and the cubical dimension.…”

“…Since we are using the product decomposition theorem to gain arbitrary gaps these manifolds will have arbitrarily large dimension. Using the results of [7] and the reflection trick of [6] we can construct 4-dimensional aspherical manifolds with arbitrary dimension gap.…”

Groups acting on CAT(0) cube complexes with uniform exponential growth

“…In the proof of Theorem B, we use a variation (Proposition 3.4) of work of Jankiewicz [Jan20] on cubical dimension of small cancellation groups (see Lemma 2.14). If instead of isolated flats, we impose the condition of hyperbolicity on our CAT(0) cube complex X, then we can use the same proof strategy as for Theorem B to relax the requirement of a geometric group action to a weakly properly discontinuous (WPD) action.…”