Let $X$ be the [classifying space](https://en.wikipedia.org/wiki/Classifying_space) of the [Higman group](https://en.wikipedia.org/wiki/Higman_group) $G$. It is well known that $G$ is an acyclic group 
$$H\_{\ast}(X;\mathbb{Z})=H\_{\ast}(pt;\mathbb{Z}).$$


Now, suppose that $\mathcal{M}$ is a [local system](https://en.wikipedia.org/wiki/Local_system) on the space $X$ such that 


$$H\_{i}(X;\mathcal{M})=0, \textrm{ for all $0\leq i$}.$$ 


Does such local system $\mathcal{M}$ on $X$ exist (other then $\mathcal{M}= 0$)



Trivial homology with local system
For $X = BG$ local systems on $X$ can be identified with $G$-modules, and homology with the derived tensor product $-\otimes^L\_{\mathbb ZG}\mathbb Z$, i.e. $H\_i(X;M) \cong \operatorname{Tor}^i\_{\mathbb Z G}(M,\mathbb Z)$. One way to see this is to take the definition $H\_i(X;M):= H\_i(\mathcal S\_\*(\widetilde X)\otimes\_{\mathbb Z\pi\_1(X)} M)$, where $\pi\_1(X)$ acts on (singular) chains on the universal cover $\widetilde X$ via deck transformations, and replace $\mathcal S\_\*(\widetilde X)$ with the cellular complex $C\_\*(\widetilde X)$ of the CW structure of the realization of the nerve of the groupoid $G//G$; then $C\_\*(\widetilde X)\otimes\_{\mathbb Z G} M = \dots\to \mathbb Z[G^2]\otimes M\to \mathbb Z[G]\otimes M\to M$ is the bar complex computing group homology.


Let $G$ be an arbitrary group, and let $M = \operatorname{ker}(\mathbb Z G\to \mathbb Z)$ be the reduced group algebra, so that we have a short exact sequence
$$
0\to M\to \mathbb ZG\to\mathbb Z\to 0
$$
of $\mathbb ZG$-modules. This gives rise to a long exact sequence of Tor-groups, in particular a boundary operator $\partial:\operatorname{Tor}^{i+1}\_{\mathbb Z G}(\mathbb Z,\mathbb Z)\to \operatorname{Tor}^i\_{\mathbb Z G}(M,\mathbb Z)$. Its kernel is the image of the map $0 = \operatorname{Tor}^{i+1}\_{\mathbb Z G}(\mathbb Z G,\mathbb Z)\to \operatorname{Tor}^{i+1}\_{\mathbb Z G}(\mathbb Z,\mathbb Z)$, so it is always injective; its cokernel is the kernel of the map $\operatorname{Tor}^{i}\_{\mathbb Z G}(\mathbb Z G,\mathbb Z)\to \operatorname{Tor}^{i}\_{\mathbb Z G}(\mathbb Z,\mathbb Z)$, which is an isomorphism for $i = 0$ and has the zero group as its codomain for $i > 0$, so it is always injective, so $\partial$ is always surjective and thus always an isomorphism.


In your example of the Higman group, we know that $H\_i(X;\mathbb Z)$ is $\mathbb Z$ concentrated in degree $0$, so that $\partial$ is an isomorphism with the zero group, so that $H\_i(X;M) = 0$ for all $i \ge 0$.


