Submission date: 25 October 2020

Abstract:

We lay the foundations of the first-order model theory of Coxeter groups. Firstly, with the exception of the 2-spherical non-affine case (which we leave open), we characterize the superstable Coxeter groups of finite rank, which we show to be essentially the Coxeter groups of affine type. Secondly, we characterize the Coxeter groups of finite rank which are domains, a central assumption in the theory of algebraic geometry over groups, which in many respects (e.g. λ-stability) reduces the model theory of a given Coxeter system to the model theory of its associated irreducible components. In the second part of the paper we move to specific definability questions in right-angled Coxeter groups (RACGs) and 2-spherical Coxeter groups. In this respect, firstly, we prove that RACGs of finite rank do not have proper elementary subgroups which are Coxeter groups, and prove further that reflection independent ones do not have proper elementary subgroups at all. Secondly, we prove that if (W, S) is a right-angled Coxeter system of finite rank, then W is a prime model of its theory if and only if the monoid Sim(W, S) of S-self-similarities of W is finitely generated, and use this to prove that if W is a universal Coxeter group of finite rank at least two, then Th(W) does not have a prime model. Thirdly, we prove that in reflection independent right-angled Coxeter groups of finite rank the Coxeter elements are type-determined. We then move to 2-spherical Coxeter groups, proving that if (W, S) is irreducible, 2-spherical even and not affine, then W is a prime model of its theory, and that if W_Γ and W_Θ are as in the previous sentence, then W_Γ is elementary equivalent to W_Θ if and only if Γ ≅ Θ, thus solving the elementary equivalence problem for most of the 2-spherical Coxeter groups.

Mathematics Subject Classification: 03C45, 03C68, 20F55, 51F15

Keywords and phrases:

Full text arXiv 2010.13161: pdf, ps.