Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 134 / 3 / d / Solution

Because $X$ is special, the fundamental group of a special cube complex embeds in a Right-angled Artin group. Right-angled Artin groups are residually finite by the residual finiteness of a right-angled Artin group, and a subgroup of a residually finite group is residually finite. Hence ${\pi }_{1}X$ is residually finite.

If ${\pi }_{1}X$ were simple, choose $1\ne g\in {\pi }_{1}X$. A finite quotient in which $g$ survives has a proper normal kernel. Simplicity would force that kernel to be trivial, embedding ${\pi }_{1}X$ into a finite group, contrary to the assumption that ${\pi }_{1}X$ is infinite. This is precisely the obstruction that an infinite residually finite group is not simple.