Is there a nontrivial example of an acyclic group $G$ such that its corresponding Eilenberg space $K(G,1)$ is homotopy equivalent to a finite CW-complex ? 



Acyclic group and finite CW-complex
The Higman group with presentation
$$\langle{a,b,c,d}\mid{aba^{-1}b^{-2}},~bcb^{-1}c^{-2},~cdc^{-1}d^{-2},~
dad^{-1}a^{-2}\rangle$$
is perfect, and the 2-complex associated to this presentation
has Euler characteristic 0. Hence this complex is acyclic.
It is in fact aspherical, but it may be simpler to observe that
Higman's group is also an iterated generalized free product with 
amalgamations $(A\*\_{\langle{b}\rangle}B)\*\_{F(a,c)}(C\*\_{\langle{c}\rangle}D)$, where $A,B,C$ and $D$ are copies of the Baumslag-Solitar group $BS(1,2)$, generated by $\{a,b\}$, $\{b,c\}$, $\{c,d\}$ and $\{d,a\}$, respectively. We may assemble a 2-dimensional Eilenberg-Mac Lane complex for the Higman group in a similar way.


