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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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