Background an objectives: Our recent work has focused on the application of infinite group theory and related algebraic geometric tools in the context of transcription factors and microRNAs. We were able to differentiate between “healthy” nucleotide sequences and disrupted sequences that may be associated with various diseases. In this paper, we extend our efforts to the study of messenger RNA (mRNA) metabolism, showcasing the power of our approach. Methods: To achieve this, we used: (a) infinite (finitely generated) groups , with generators representing the distinct nucleotides and a relation between them [e.g., the consensus sequence in the mRNA translation (i), the poly(A) tail in item (ii), and the microRNA seed in item (iii)]; (b) aperiodicity theory, which connects healthy groups to free groups of rank r and their profinite completion , and (c) the representation theory of groups over the space-time-spin group SL2(C), highlighting the role of surfaces with isolated singularities in the character variety. Results: We investigate (1) mRNA translation in prokaryotes and eukaryotes, (2) polyadenylation in eukaryotes, which is crucial for nuclear export, translation, stability, and splicing of mRNA, (3) microRNAs involved in RNA silencing and post-transcriptional regulation of gene expression, and (4) identification of disrupted sequences that could lead to potential illnesses. Conclusion: Our approach could potentially contribute to the understanding of the molecular mechanisms underlying various diseases and help develop new diagnostic or therapeutic strategies.