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arXiv · 2608.10150

Quasirandomness and Uniform Twin-Width

George Kontogeorgiou·Bobby Miraftab
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Abstract

For every nontrivial finite group, we prove that its quasirandom degree gives a polynomial lower bound on its uniform twin-width, whereas its minimum faithful complex representation degree gives a linear upper bound. For nonabelian finite simple groups, these two parameters coincide, so uniform twin-width is polynomially equivalent to quasirandomness in that class, yielding a new definition of quasirandomness in the sense of Gowers. We use the lower bound to prove that uniform twin-width is unbounded over finite groups, which helps us construct finitely presented groups with finite twin-width but infinite uniform twin-width. This answers a question of Bonnet, Geniet, Tessera and Thomasse. Finally, we determine the uniform twin-width of all three Thompson groups.

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George Kontogeorgiou, Bobby Miraftab. 2026-08-10. Quasirandomness and Uniform Twin-Width. https://arxiv.org/abs/2608.10150

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