The 2-Sheeted, 3-Sheeted, and Universal Coverings of Corresponding 2-Oriented Graph of Rank-2 Free Group

Abstract

This paper reviews results from classical covering theory and applies the classification theorem to provide a comprehensive classification of the connected 2-sheeted and 3-sheeted covering spaces of the bouquet of 2 circles, up to isomorphism without basepoints. These coverings are represented as 4-valent graphs, labeled by the two generators of the rank-2 free group, which serves as the fundamental group of the bouquet of 2 circles. The result directly addresses a question posed by Hatcher, a problem often answered incompletely in previous attempts. This paper provides all possible cases and includes detailed illustrations to accompany the classifications. Additionally, the paper explores the X-digraphs associated with the corresponding index-2 and index-3 subgroups of the rank-2 free group. These subgroup graphs introduce powerful combinatorial tools to examine the subgroups of free groups. In the broader context of geometric group theory, the paper concludes by discussing the Cayley graph of the rank-2 free group as a universal covering space.