A uniform Tits alternative for endomorphisms of the projective line

Alonso Beaumont

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arXiv · 2504.14263

A uniform Tits alternative for endomorphisms of the projective line

Alonso Beaumont

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Abstract

A recent article of J.P. Bell, K. Huang, W. Peng and T.J. Tucker establishes an analog of the Tits alternative for semigroups of endomorphisms of the projective line. The proof involves a ping-pong argument on arithmetic height functions. Extending this method, we obtain a uniform version of the same alternative. In particular, we show that semigroups of $\mathrm{End}(\mathbb{P}^{1})$ of exponential growth are of uniform exponential growth.

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Alonso Beaumont. 2025-04-19. A uniform Tits alternative for endomorphisms of the projective line. https://arxiv.org/abs/2504.14263

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