2018/10/26 by Simon André, André, Simon
Computer Science · Mathematics · #FOS: Mathematics #Geometric and Algebraic Topology #Group Theory (math.GR) #Logic (math.LO) #Mathematical Dynamics and Fractals #semigroups and automata theory
paper · pdf · doi:10.48550/arxiv.1810.11200
openalex publication_date 2018/10/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Free groups are known to be homogeneous, meaning that finite tuples of elements which satisfy the same first-order properties are in the same orbit under the action of the automorphism group. We show that virtually free groups have a slightly weaker property, which we call uniform almost-homogeneity: the set of k-tuples which satisfy the same first-order properties as a given k-tuple u is the union of a finite number of Aut(G)-orbits, and this number is bounded independently from u and k. Moreover, we prove that there exists a virtually free group which is not ∃-homogeneous. We also prove that all hyperbolic groups are homogeneous in a probabilistic sense.