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## NILPOTENT GROUPS

*John C. Lennox and Derek J. S. Robinson*

### in The Theory of Infinite Soluble Groups

- Published in print:
- 2004
- Published Online:
- September 2007
- ISBN:
- 9780198507284
- eISBN:
- 9780191709326
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/acprof:oso/9780198507284.003.0002
- Subject:
- Mathematics, Pure Mathematics

This chapter discusses extraction of roots in nilpotent groups. Topics covered include the Mal’cev completion for a torsion-free nilpotent group, the commutator collection process, and isolators.

## Statement E Implies Statement D

*Ben Brubaker, Daniel Bump, and Solomon Friedberg*

### in Weyl Group Multiple Dirichlet Series: Type A Combinatorial Theory (AM-175)

- Published in print:
- 2011
- Published Online:
- October 2017
- ISBN:
- 9780691150659
- eISBN:
- 9781400838998
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691150659.003.0014
- Subject:
- Mathematics, Combinatorics / Graph Theory / Discrete Mathematics

This chapter introduces the theorem that says Statement E implies Statement D, first by fixing a nodal signature η and presenting a “cut and paste” virtual resotope. It then presents the proof, ... More

## Office Hours with a Geometric Group Theorist

*Dan Margalit and Matt Clay (eds)*

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- book

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.001.0001
- Subject:
- Mathematics, Geometry / Topology

Geometric group theory is the study of the interplay between groups and the spaces they act on, and has its roots in the works of Henri Poincaré, Felix Klein, J.H.C. Whitehead, and Max Dehn. This ... More

## Groups Acting on Trees

*Dan Margalit*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0003
- Subject:
- Mathematics, Geometry / Topology

This chapter considers groups acting on trees. It examines which groups act on which spaces and, if a group does act on a space, what it says about the group. These spaces are called trees—that is, ... More

## Right-Angled Artin Groups

*Robert W. Bell and Matt Clay*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0014
- Subject:
- Mathematics, Geometry / Topology

This chapter deals with right-angled Artin groups, a broad spectrum of groups that includes free groups on one end, free abelian groups on the other end, and many other interesting groups in between. ... More

## Free Groups and Folding

*Matt Clay*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0004
- Subject:
- Mathematics, Geometry / Topology

This chapter studies subgroups of free groups using the combinatorics of graphs and a simple operation called folding. It introduces a topological model for free groups and uses this model to show ... More

## The Ping-Pong Lemma

*Johanna Mangahas*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0005
- Subject:
- Mathematics, Geometry / Topology

This chapter considers an identifying feature of free groups: their ability to play ping-pong. In mathematics, you may encounter a group without immediately knowing which group it is. Fortunately, ... More

## Automorphisms of Free Groups

*Matt Clay*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0006
- Subject:
- Mathematics, Geometry / Topology

This chapter discusses the automorphisms of free groups. Every group is the collection of symmetries of some object, namely, its Cayley graph. A symmetry of a group is called an automorphism; it is ... More

## Asymptotic Dimension

*Greg Bell*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0011
- Subject:
- Mathematics, Geometry / Topology

This chapter considers the notion of asymptotic dimension. It first provides an overview of asymptotic dimension before explaining the asymptotic dimensions of free abelian and nonabelian groups, ... More

## Basics

*Christophe Reutenauer*

### in From Christoffel Words to Markoff Numbers

- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.003.0002
- Subject:
- Mathematics, Pure Mathematics

Definitions and basic results about words: alphabet, length, free monoid, concatenation, prefix, suffix, factor, conjugation, reversal, palindrome, commutative image, periodicity, ultimate ... More

## Ends of Groups

*Nic Koban and John Meier*

### in Office Hours with a Geometric Group Theorist

- Published in print:
- 2017
- Published Online:
- May 2018
- ISBN:
- 9780691158662
- eISBN:
- 9781400885398
- Item type:
- chapter

- Publisher:
- Princeton University Press
- DOI:
- 10.23943/princeton/9780691158662.003.0010
- Subject:
- Mathematics, Geometry / Topology

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 ... More

## Two Flavours of Mathematical Explanation

*Mark Colyvan, John Cusbert, and Kelvin McQueen*

### in Explanation Beyond Causation: Philosophical Perspectives on Non-Causal Explanations

- Published in print:
- 2018
- Published Online:
- June 2018
- ISBN:
- 9780198777946
- eISBN:
- 9780191823404
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198777946.003.0012
- Subject:
- Philosophy, Philosophy of Science, Metaphysics/Epistemology

A proof of a mathematical theorem tells us that the theorem is true (or should be accepted), but some proofs go further and tell us why the theorem is true (or should be accepted). That is, some, but ... More

## Bases and Automorphisms of the Free Group on Two Generators

*Christophe Reutenauer*

### in From Christoffel Words to Markoff Numbers

- Published in print:
- 2018
- Published Online:
- January 2019
- ISBN:
- 9780198827542
- eISBN:
- 9780191866418
- Item type:
- chapter

- Publisher:
- Oxford University Press
- DOI:
- 10.1093/oso/9780198827542.003.0017
- Subject:
- Mathematics, Pure Mathematics

The chapter begins with a self-contained exposition of the theory of Nielsen on the free groupwith two generators: bases of F(a, b),Nielsen’s criterion for automorphisms of F(a, b), It also ... More

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