The intersection of formal language theory and group theory provides a fascinating lens for studying algebraic structures, most notably through the word problem. This connection has allowed mathematicians to classify groups based on the Chomsky hierarchy. While the landmark Muller-Schupp theorem completely characterized groups with context-free word problems, progressing beyond this boundary has remained a major challenge. In this talk, we shift toward a more combinatorial and geometric approach. We will introduce a new framework that yields a characterization of the f.g. subgroups of Thompson V. Notably, this group is central to a well-known 2008 conjecture regarding groups with co-context-free word problems. Joint works with Corentin Bodart, Daniele D’Angeli, Francesco Matucci and Emanuele Rodaro.