3 Stallings’ automata
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Jordi Delgado
Abstract
This chapter provides a self-contained reference for Stallings’ automata theory, which allows to interpret every subgroup of the free group as a neat geometric object: its Stallings automaton. In contrast to the original topological approach-rooted in the seminal works of Serre, Stallings, and others-we adopt a discrete perspective, where geometric objects are abstract combinatorial structures, and subsequently reveal the connections with the underlying topological notions. This allows us to highlight both sides of the theory: the algebraic part (closely linked to the topological point of view), and the computational part (more naturally described in terms of automata and language theory). Once the bijection between subgroups and automata is established, we focus on some classical applications of this theory. From the algebraic point of view, we deduce the Nielsen-Schreier theorem, the Schreier index formula, Howson’s property for free groups (along with classical bounds for the rank of the intersection of subgroups), Marshall Hall’s theorem, and properties such as Hopfianity and residual finiteness of free groups. On the other hand, the computability of the Stallings automaton for finitely generated subgroups, based on the now classical technique of Stallings foldings, yields a plethora of algorithmic results. We prove the computability of the subgroup membership problem, the finite index problem, the subgroup conjugacy problem, the coset intersection problem; as well as the computability of intersections of finitely generated subgroups, and the set of algebraic extensions of a finitely generated subgroup, among other results.
Abstract
This chapter provides a self-contained reference for Stallings’ automata theory, which allows to interpret every subgroup of the free group as a neat geometric object: its Stallings automaton. In contrast to the original topological approach-rooted in the seminal works of Serre, Stallings, and others-we adopt a discrete perspective, where geometric objects are abstract combinatorial structures, and subsequently reveal the connections with the underlying topological notions. This allows us to highlight both sides of the theory: the algebraic part (closely linked to the topological point of view), and the computational part (more naturally described in terms of automata and language theory). Once the bijection between subgroups and automata is established, we focus on some classical applications of this theory. From the algebraic point of view, we deduce the Nielsen-Schreier theorem, the Schreier index formula, Howson’s property for free groups (along with classical bounds for the rank of the intersection of subgroups), Marshall Hall’s theorem, and properties such as Hopfianity and residual finiteness of free groups. On the other hand, the computability of the Stallings automaton for finitely generated subgroups, based on the now classical technique of Stallings foldings, yields a plethora of algorithmic results. We prove the computability of the subgroup membership problem, the finite index problem, the subgroup conjugacy problem, the coset intersection problem; as well as the computability of intersections of finitely generated subgroups, and the set of algebraic extensions of a finitely generated subgroup, among other results.
Chapters in this book
- Frontmatter I
- Contents V
- List of Contributing Authors VII
- 1 Cellular automata over shifts and subshifts: Garden of Eden theorems and surjunctivity 1
- 2 Languages, groups, and equations 63
- 3 Stallings’ automata 95
- 4 Groups with multiple context-free word problems 161
- 5 Parallel complexity in group theory 183
- 6 The word problem for automaton groups 265
- Index 397
Chapters in this book
- Frontmatter I
- Contents V
- List of Contributing Authors VII
- 1 Cellular automata over shifts and subshifts: Garden of Eden theorems and surjunctivity 1
- 2 Languages, groups, and equations 63
- 3 Stallings’ automata 95
- 4 Groups with multiple context-free word problems 161
- 5 Parallel complexity in group theory 183
- 6 The word problem for automaton groups 265
- Index 397
