Equality G^{0} = G^{00} and virtual algebraicity for definable groups over C((t))

Prove that for every group G definable in the valued field C((t)), the definable connected component G^{0} equals the smallest type-definable subgroup G^{00} of bounded index; consequently, show that G has a finite-index subgroup that is an open subgroup of an algebraic group over C((t)).

Background

The relationship between G^{0} and G^{00} is central in the model-theoretic analysis of definable groups, often driving structural classification and connections to algebraic groups. In p-adically closed fields, equality results lead to strong algebraicity consequences (e.g., [7]).

The authors conjecture that the same phenomenon holds over C((t)), implying that definable groups are virtually algebraic in the sense of containing a finite-index subgroup that is open in an algebraic group over C((t)).