# Acyclic group and finite CW-complex

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 ?

• It would be helpful to say what you mean by "nontrivial". If it means "nontrivial group" I would move the adjective accordingly. – YCor Feb 5 at 22:46

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.
I presume by "acyclic" you are referring to homology with $$\mathbb{Z}$$ coefficients. There are many such examples.
For instance, you can take two elements $$u,v$$ in the free group $$F_2$$ of rank 2 that satisfy the $$C'(1/6)$$ small-cancellation condition, and also such that $$u,v$$ together generate the abelianisation $$\mathbb{Z}^2$$. Explicit examples are easy to construct.
The small-cancellation condition then implies that the corresponding presentation complex $$X$$ is aspherical, and the assumption about the abelianisation implies that $$X$$ has the homology of a point. But $$\pi_1X$$ is an infinite hyperbolic group, in particular non-trivial.
• @AndréHenriques Sure, take $F_2 = \langle a,b\rangle$. A suitable pair of elements is $a(aba^{-1}b^{-1})^5$ and $b^6a^2b^3a^{-1}b^{-8}a^{-1}$. I believe the longest common piece is $a^2b$, and $3$ is less than one sixth of the length of each word, which is cyclically reduced. Counting exponent sums, we see that they generate the abelianization. – Rylee Lyman Feb 6 at 3:39