Definably compact groups over C((t)) are generically stable and have G^{00} = G^{0} of finite index

Establish that for every definably compact group G definable over the valued field C((t)) in the language of rings: (i) G is generically stable (i.e., admits a generically stable global type), and (ii) the smallest type-definable subgroup of G of bounded index, denoted G^{00}, equals the definable connected component G^{0}, and this subgroup has finite index in G.

Background

Section 3 develops a weak Lie structure on definable groups over C((t)), enabling a definable topology and the notion of definable compactness for such groups via finitely many affine pieces. It also establishes that fsg groups are definably compact, paralleling results in p-adically closed fields.

Within this framework, the authors propose a revised version of a conjecture from Ling–Yao [1], asserting stronger structural properties of definably compact groups over C((t)), namely generic stability and the identification of G^{00} with G^{0} as a finite-index subgroup.