ON COMBINATORIAL STRUCTURES ARISING IN GROUP THEORY

THIS THESIS EXPLORES TWO CENTRAL THEMES IN CONTEMPORARY GROUP THEORY. THE FIRST CONCERNS GRAPHS NATURALLY ARISING FROM GROUPS, SHOWING HOW THEIR COMBINATORIAL FEATURES CAN ILLUMINATE SIGNIFICANT ASPECTS OF THE GROUP’S ALGEBRAIC STRUCTURE. THE SECOND ADDRESSES EMBEDDING PROBLEMS, WITH A FOCUS ON VERBAL SUBGROUPS AND SUBGROUPS DEFINED BY COMMUTATOR CONDITIONS, HIGHLIGHTING HOW SUCH EMBEDDINGS REFLECT DEEPER STRUCTURAL CONSTRAINTS. TOGETHER, THESE PERSPECTIVES ILLUSTRATE THE INTERPLAY BETWEEN COMBINATORIAL METHODS AND ALGEBRAIC PROPERTIES IN THE STUDY OF GROUPS.