Skip to main content Contents Index Prev Up Next \(\DeclareMathOperator{\ann}{ann}
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Chapter 4 Presentations, Free Groups (*)
Coming soon!
Question 4.1 . The What, Where, and Why of Group Presentations.
Why are group presentations important? Where are they found? What are they used for?
Answer .
Understanding Module Presentations. The study of group presentations helps understand the more general concept of presentations in algebraic structures, including modules. Presentations of modules are essential in the classification of finitely generated modules over an PID, which we will see CITEX. Specifically, we can use Smith Normal Form, a process that simplifies the presentation of a module by transforming the relation matrix into a diagonal form, revealing the structure of the module in terms of its free and torsion parts.
Combinatorial Group Theory. Group presentations are the cornerstone of combinatorial group theory, where groups are studied through their generators and relations. For example, the word problem asks whether two words (representing elements of the group) are equivalent under the relations given by the group presentation. The difficulty of solving this problem depends on the complexity of the presentation.
Geometric Group Theory. The Cayley graph of a group is a graph that visually represents the group’s structure in terms of its generators. The group’s presentation defines the edges of the graph, with each generator corresponding to an edge between vertices (group elements). This graph-theoretic representation is central to many areas of geometric group theory, such as the study of word-hyperbolic groups.
Algebraic Topology. In algebraic topology, group presentations are used to describe the fundamental group \(π_1(X)\) of a topological space \(X\text{,}\) which encodes information about the loops in the space up to homotopy. For example, Given a CW complex (a type of topological space built from cells), the fundamental group of the space can be described using a group presentation. The generators correspond to the \(1\) -cells (edges), and the relations come from the \(2\) -cells (faces), which describe how the edges are glued together.
Knot Theory. The knot group of a knot \(K\) in \(3\) -dimensional space is the fundamental group of the knot complement (the space obtained by removing the knot from \(\R^3\) ). This group is often described using a group presentation.
Algebraic Geometry and Algebraic Groups. The fundamental group of an algebraic variety can often be computed using a group presentation, especially for varieties with a well-understood topological structure (e.g., plane curves or higher-dimensional varieties).
Theoretical Computer Science. Group presentations are connected to automata theory through the study of automatic groups, where the word problem can be solved efficiently. Groups with certain types of presentations can be recognized by finite automata, and this has applications in both algebra and computer science.
