Logic Seminar
John Baldwin
UIC
Categoricity of covers of non-arithmetic Fuchsian groups
Abstract: We sketch the proof that if a Fuchsian group $\Gamma$ has finite index in its commensurator (i.e. is non-arithmetical), then the
theory of its two sorted universal cover structure is axiomatized by a sentence in $L_{\omega_1,\omega}$ that
is categorical in uncountable powers. This is a simpler version of the earlier result in the arithmetic
case (Daw & Harris; Eterović). The argument is structured rather differently and clarifies the roles of model theory and
geometry in the earlier case by replacing some `geometric/number theoretic' arguments by model theoretic ones. We point to the
role of finite index in allowing these simplifications. Joint work with Joel Nagloo and incorporating some earlier work
with Andres Villaveces in a survey of Zilber's `logically perfect structures are $L_{\omega_1,\omega}(Q)$-categorical in
uncountable power.
Tuesday November 19, 2024 at 4:00 PM in 636 SEO