Ends of Groups

This chapter focuses on the ends of a group. It first constructs a group action on the Cantor set and creates a free group from bijections of the Cantor set before showing how the idea of trying to understand what is happening at infinity for an infinite group is captured by the phrase “the ends of a group.” It then explores the notion of ends in the context of infinite graphs and presents examples that provide some insight into the number of ends of groups. It also looks at semidirect products and demonstrates how to calculate the number of ends of the braid groups before moving beyond the process of counting the ends of a group, taking into account the ends of the 4-valent tree. The discussion includes exercises and research projects.

Commensurated subgroups and ends of groups

IfIt is straightforward to check that

Grouping Hierarchy and Trajectories of Pacing in Performances of Chopin’s Mazurkas

Theories of expressive timing in the performance of tonal music emphasize the role of grouping structure, whereby performers are understood to communicate the ends of groups through group-final lengthening (GFL). But this approach depends on a one-way mapping from a single grouping-structural analysis onto performed durations, denying a role for interpretive difference on the part of performers and analysts. Drawing on contour theory, this article reverses this mapping by presenting a method for recovering the hierarchical grouping structure of a performed phrase that is sensitive to the constraints of temporal perception. Groups whose durational contour segments reduce to a contour adjacency series of <+> or <–,+> are understood to be GFL-reflective. By observing which levels of time-span organization are GFL-reflective among different performances of the same phrase, unique construals of grouping structure can be attributed to different renditions.The article employs this method in order to examine different approaches to pacing in performances of two of Chopin’s mazurkas. The pieces in question present eight-measure themes in which the salience of different levels of grouping structure contrast. Through duration decisions, performers can accentuate, amend, or bypass these suggestions of contrast in pacing. By presenting an analytical method that recognizes the creative power of performance to interact with a grouping structure implied by a score, I hope to reshape the relationship between performers and analysts as a dialog about the possible structural descriptions a piece can support.