A sampling of remarkable groups. Thompson’s, self-similar, lamplighter, and Baumslag-Solitar.

*(English)*Zbl 1446.20001
Compact Textbooks in Mathematics. Cham: Birkhäuser (ISBN 978-3-030-01976-1/pbk; 978-3-030-01978-5/ebook). xii, 188 p. (2018).

This delightful book is dedicated to introducing an advanced undergraduate student to the study of some aspects of infinite groups theory.

The first chapter is devoted to a brief but detailed survey of basic group theory. After the strictly necessary preliminaries, the concepts of Cayley graphs of groups, of rooted \((n)\)-ary trees, of amenability, of growth of finitely generated groups and of dead-end elements are discussed.

Chapter II is dedicate to the so-called Thompson’s groups (or chameleon groups) \(\mathsf{F} < \mathsf{T} < \mathsf{V}\) introduced by Richard Thompson in some unpublished handwritten notes in 1965. The groups, \(\mathsf{T}\) and \(\mathsf{V}\) are infinite, simple, finitely presented, the first known examples of groups that enjoy these properties. The group \(\mathsf{F}\) is not simple but its derived subgroup \([\mathsf{F},\mathsf{F}]\) is and the quotient of \(\mathsf{F}\) by its derived subgroup is the free abelian group of rank 2. \(\mathsf{F}\) is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2. In the book under review, it is shown that \(\mathsf{F}\) has the remarkable presentation with two generators and two relations: \[ \mathsf{F}= \left\langle a,b \mid [ab^{-1},b^{a}]=1, \, [ab^{-1},b^{a^{2}}]=1 \right\rangle. \] Then the group \(\mathsf{F}\) is described as the set of continuous piecewise linear functions from the closed unit interval \([0,1]\) onto itself where the slope of each piece is an integer power of 2, there are a finite number of pieces, and the breakpoints in the domain between each piece are dyadic fractions. Finally, the authors show that \(\mathsf{F}\) has an alternative presentation with an infinite number of generators and relations.

In Chapter III, the authors explore groups which are called self-similar or automata groups. Elements of these groups are automorphisms of the infinite complete rooted binary tree (itself a self-similar object), and give three ways of viewing these groups, using automata, portraits, and self-similar rules. One of the most famous examples of self-similar groups is the Grigorchuk group \(\mathsf{G}\) (see [R. I. Grigorchuk, Funct. Anal. Appl. 14, 41–43 (1980; Zbl 0595.20029); translation from Funkts. Anal. Prilozh. 14, No. 1, 53–54 (1980)]). In this book, it is proved that \(\mathsf{G}\) is a 2-group finitely generated but not finitely presented and just infinite (that is \(\mathsf{G}\) is infinite but every proper quotient group of \(\mathsf{G}\) is finite). Furthermore, Grigorchuk’s group has sub-exponential growth and hence it is amenable but not elementary amenable.

The lamplighter group \(\mathsf{L}_{2}\) is the subject of Chapter IV. The group \(\mathsf{L}_{2}\) is a simple example of a wreath product, namely, \(\mathsf{L}_{2} \simeq C_{2} \wr \mathbb{Z}\). Since 1983, it has become known as the lamplighter group because it can be described as a dynamical system consisting of a doubly-infinite road with infinitely many lamps, only finitely many of which are lit, and of course a lamplighter who wanders up and down the road lighting or extinguishing individual lamps. In §4.6, it is proved that \(\mathsf{L}_{2}\) is generated by an automaton.

In Chapter V, the authors focus on a family of two-generator one-relator groups found by G. Baumslag and D. Solitar [Bull. Am. Math. Soc. 68, 199–201 (1962; Zbl 0108.02702)] given by the presentation \[\mathsf{BS}(m,n)=\left\langle a,t \mid t^{-1}a^{m}t=t^{n} \right\rangle, \qquad m,n \in \mathbb{Z} \setminus \{0\}. \] It is proved that \(\mathsf{BS}(2,3)\) is non-Hopfian (a group is called Hopfian if it is not isomorphic to any of its proper quotients). Moreover, it is proved that \(\mathsf{BS}(1,n)\) is Hopfian and that \(\mathsf{BS}(12,18)\) is virtually non-Hopfian because it has a normal subgroup of finite index \(N\) such that \(\mathsf{BS}(12,18)/N \simeq \mathsf{BS}(2,3)\).

A theorem of A. M. Brunner and S. Sidki [Int. J. Algebra Comput. 8, No. 1, 127–139 (1998; Zbl 0923.20023)] establishes that a linear group over \(\mathbb{Z}\) can be represented by a finite-state automaton using a four letter alphabet. Theorem 5.2 states that \(\mathsf{BS}(m,m)\) is linear over \(\mathbb{Z}\), and hence self-similar. The above mentioned result provides a link between Baumslag-Solitar groups and the topics covered in Chapter III.

The final chapter contains detailed and extensive solutions to of the exercises provided in the text. The whole book is enriched with beautiful and instructive illustrations.

The first chapter is devoted to a brief but detailed survey of basic group theory. After the strictly necessary preliminaries, the concepts of Cayley graphs of groups, of rooted \((n)\)-ary trees, of amenability, of growth of finitely generated groups and of dead-end elements are discussed.

Chapter II is dedicate to the so-called Thompson’s groups (or chameleon groups) \(\mathsf{F} < \mathsf{T} < \mathsf{V}\) introduced by Richard Thompson in some unpublished handwritten notes in 1965. The groups, \(\mathsf{T}\) and \(\mathsf{V}\) are infinite, simple, finitely presented, the first known examples of groups that enjoy these properties. The group \(\mathsf{F}\) is not simple but its derived subgroup \([\mathsf{F},\mathsf{F}]\) is and the quotient of \(\mathsf{F}\) by its derived subgroup is the free abelian group of rank 2. \(\mathsf{F}\) is totally ordered, has exponential growth, and does not contain a subgroup isomorphic to the free group of rank 2. In the book under review, it is shown that \(\mathsf{F}\) has the remarkable presentation with two generators and two relations: \[ \mathsf{F}= \left\langle a,b \mid [ab^{-1},b^{a}]=1, \, [ab^{-1},b^{a^{2}}]=1 \right\rangle. \] Then the group \(\mathsf{F}\) is described as the set of continuous piecewise linear functions from the closed unit interval \([0,1]\) onto itself where the slope of each piece is an integer power of 2, there are a finite number of pieces, and the breakpoints in the domain between each piece are dyadic fractions. Finally, the authors show that \(\mathsf{F}\) has an alternative presentation with an infinite number of generators and relations.

In Chapter III, the authors explore groups which are called self-similar or automata groups. Elements of these groups are automorphisms of the infinite complete rooted binary tree (itself a self-similar object), and give three ways of viewing these groups, using automata, portraits, and self-similar rules. One of the most famous examples of self-similar groups is the Grigorchuk group \(\mathsf{G}\) (see [R. I. Grigorchuk, Funct. Anal. Appl. 14, 41–43 (1980; Zbl 0595.20029); translation from Funkts. Anal. Prilozh. 14, No. 1, 53–54 (1980)]). In this book, it is proved that \(\mathsf{G}\) is a 2-group finitely generated but not finitely presented and just infinite (that is \(\mathsf{G}\) is infinite but every proper quotient group of \(\mathsf{G}\) is finite). Furthermore, Grigorchuk’s group has sub-exponential growth and hence it is amenable but not elementary amenable.

The lamplighter group \(\mathsf{L}_{2}\) is the subject of Chapter IV. The group \(\mathsf{L}_{2}\) is a simple example of a wreath product, namely, \(\mathsf{L}_{2} \simeq C_{2} \wr \mathbb{Z}\). Since 1983, it has become known as the lamplighter group because it can be described as a dynamical system consisting of a doubly-infinite road with infinitely many lamps, only finitely many of which are lit, and of course a lamplighter who wanders up and down the road lighting or extinguishing individual lamps. In §4.6, it is proved that \(\mathsf{L}_{2}\) is generated by an automaton.

In Chapter V, the authors focus on a family of two-generator one-relator groups found by G. Baumslag and D. Solitar [Bull. Am. Math. Soc. 68, 199–201 (1962; Zbl 0108.02702)] given by the presentation \[\mathsf{BS}(m,n)=\left\langle a,t \mid t^{-1}a^{m}t=t^{n} \right\rangle, \qquad m,n \in \mathbb{Z} \setminus \{0\}. \] It is proved that \(\mathsf{BS}(2,3)\) is non-Hopfian (a group is called Hopfian if it is not isomorphic to any of its proper quotients). Moreover, it is proved that \(\mathsf{BS}(1,n)\) is Hopfian and that \(\mathsf{BS}(12,18)\) is virtually non-Hopfian because it has a normal subgroup of finite index \(N\) such that \(\mathsf{BS}(12,18)/N \simeq \mathsf{BS}(2,3)\).

A theorem of A. M. Brunner and S. Sidki [Int. J. Algebra Comput. 8, No. 1, 127–139 (1998; Zbl 0923.20023)] establishes that a linear group over \(\mathbb{Z}\) can be represented by a finite-state automaton using a four letter alphabet. Theorem 5.2 states that \(\mathsf{BS}(m,m)\) is linear over \(\mathbb{Z}\), and hence self-similar. The above mentioned result provides a link between Baumslag-Solitar groups and the topics covered in Chapter III.

The final chapter contains detailed and extensive solutions to of the exercises provided in the text. The whole book is enriched with beautiful and instructive illustrations.

Reviewer: Enrico Jabara (Venezia)

##### MSC:

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |

20E06 | Free products of groups, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations |

20E08 | Groups acting on trees |

20E22 | Extensions, wreath products, and other compositions of groups |

20F10 | Word problems, other decision problems, connections with logic and automata (group-theoretic aspects) |

20F50 | Periodic groups; locally finite groups |

20F65 | Geometric group theory |

22F50 | Groups as automorphisms of other structures |