On the model theory of the Farey graph
Zahra Mohammadi Khangheshlaghi, Katrin Tent
TL;DR
This work axiomatizes the Farey graph's theory Th(G_F) using two geometric/combinatorial conditions and proves its $ω$-stability with Morley rank $ω$. The authors construct the Farey graph as a Hrushovski limit of a finite-graph class with strong amalgamation, and they develop an expanded language to achieve quantifier elimination. They compute Morley ranks for definable families and establish a forking independence framework akin to known results in related geometries, highlighting deep connections between model theory and geometric group theory. The results bridge combinatorial graph constructions with stability-theoretic analysis, contributing to the understanding of the free factor complex from a model-theoretic perspective.
Abstract
We axiomatize the theory of the Farey graph and prove that it is $ω$-stable of Morley rank $ω$.