Good fundamental groups for the disk-embedding theorem

Let $\mathrm{\Gamma }$ be a group. Say that $\mathrm{\Gamma }$ is good for the Freedman disk-embedding theorem when it has the ${\pi }_1$-null disk property: for every disk-like capped grope $G^c$ of height $1.5$ with a chosen basepoint and every homomorphism $\varphi :{\pi }_1(G^c)\to \mathrm{\Gamma }$, there is a proper framed generic immersion $d:D^2\looparrowright G^c$ of a disk into the grope's actual four-dimensional framed block. Its boundary is the core attaching circle, with the normal framing specified by the attaching solid-torus neighborhood, and each double-point loop of $d$, regarded after basing as an element of ${\pi }_1(G^c)$, is sent to the identity by $\varphi$. This definition of a good group is a hypothesis on $\mathrm{\Gamma }$; it does not assert that any particular group is good.

The precise 4-dimensional grope, height, attaching-region, and double-point-loop conventions

Stefan Behrens et al., The Disc Embedding Theorem, Definition 12.12 and Chapter 19, https://dokumen.pub/the-disc-embedding-theorem-0198841310-9780198841319.html