Soficity, short cycles and the Higman group

This is a paper with two aims. First, we show that the map from $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ to itself defined by exponentiation $x\to m^x$ has few $3$-cycles – that is to say, the number of cycles of length three is $o(p)$. This improves on previous bounds. Our second objective is to contribute to an ongoing discussion on how to find a non-sofic group. In particular, we show that, if the Higman group were sofic, there would be a map from $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ to itself, locally like an exponential map, yet satisfying a recurrence property.