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arXiv · 2606.02225

On prime endomorphisms of the free group of rank two

Andreas Thom
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Abstract

We study prime endomorphisms of the free group $F_2$. The main results provide lower and upper bounds for the logarithmic density $\delta_{\mathrm{np}}$ of non-prime endomorphisms: $$ \frac34 \leq \delta_{\mathrm{np}}\leq \frac{27}{28}. $$ Equivalently, we prove corresponding lower and upper bounds for the exponential decay of the probability that a random endomorphism is non-prime, aligning with a heuristic argument of Ian Agol.

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Andreas Thom. 2026-06-01. On prime endomorphisms of the free group of rank two. https://arxiv.org/abs/2606.02225

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