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arXiv · 2209.04069

Free structures and limiting density

Johanna N. Y. Franklin·Meng-Che "Turbo" Ho·Julia Knight
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Abstract

Gromov asked what a typical (finitely presented) group looks like, and he suggested a way to make the question precise in terms of limiting density. The typical finitely generated group is known to share some important properties with the non-abelian free groups. We ask Gromov's question more generally, for structures in an arbitrary algebraic variety (in the sense of universal algebra), with presentations of a specific form. We focus on elementary properties. We give examples illustrating different behaviors of the limiting density. Based on the examples, we identify sufficient conditions for the elementary first-order theory of the free structure to match that of the typical structure; i.e., a sentence is true in the free structure iff it has limiting density 1.

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BibTeXRIS

Johanna N. Y. Franklin, Meng-Che "Turbo" Ho, Julia Knight. 2022-09-09. Free structures and limiting density. https://arxiv.org/abs/2209.04069

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