If the deficiency of a presentation $P$ is $0$ and $P$ is aspherical, then the deficiency of the group $P$ defines is $0$.

Question

I need a reference for the following theorem:

If the deficiency of a presentation $P$ is $0$ and $P$ is aspherical, then the deficiency of the group $P$ defines is also $0$.

I think it's by Trotter but I haven't found a paper on it.

Please help :)

Answer 1

The following article might serve as a reference:

Epstein, D. B. A. Finite presentations of groups and 3-manifolds. Quart. J. Math. Oxford Ser. (2) 12 1961 205–212.

Here is a proof of the statement that $d(G) = d(P)$ when the presentation 2-complex $X$ of $P$ is aspherical and $G$ is isomorphic to the group defined by the finite presentation $P$.

Lemma 1.2 of the article cited says that for any finite presentation $P$ of $G$, we have that

$$d(P) \leq {\rm rank \,} H_1(G;\mathbb{Z}) - s(H_2(G;\mathbb{Z})),$$

where for a group $A$, $s(A)$ is the minimal number of generators.

Now consider any finite presentation $P$ of $G$ and $Q$ a finite presentation of $G$ with an aspherical presentation 2-complex $X$. The homology of $G$ is that of $X$. Since $X$ is 2-dimensional, $H_2(X,\mathbb{Z})$ is free abelian and so $s(H_2(G;\mathbb{Z}) = {\rm rank \,}H_2(G;\mathbb{Z})$. Thus, Epstein's bound becomes

$$d(P) \leq 1 - \chi(X) = d(Q).$$

Therefore, $d(Q)$ has maximal deficiency among all finite presentations for $G$. Notice that we do not need to assume that the deficiency is zero.