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Let $$\Gamma $$ be a finite simplicial graph such that the flag complex on $$\Gamma $$ is a 2-dimensional triangulated disk. We show that with some assumptions, the Dehn function of the associated Bestvina–Brady group is either quadratic, cubic, or quartic. Furthermore, we can identify the Dehn function from the defining graph $$\Gamma $$ .

Baumslag’s group is a finitely presented metabelian group with a \({\mathbb Z \wr \mathbb Z}\) subgroup. There is an analogue with an additional torsion relation in which this subgroup becomes \({C_m \wr \mathbb Z}\). We prove that Baumslag’s group has an exponential Dehn function. This contrasts with the torsion analogues which have quadratic Dehn functions.

We show that PSL2(Z[1/p]) admits a combing with bounded asynchronous width, and use this combing to show that PSL2(Z[1/p]) has an exponential Dehn function. As a corollary, PSL2(Z[1/p]) has solvable word problem and is not an automatic group.