From Christoffel words to braids

From Christoffel words to braids

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From Christoffel words to braids Christian Kassel Institut de Recherche Mathématique Avancée CNRS - Université Louis Pasteur Strasbourg, France Colloquium at Rutgers 13 April 2007

matematica pura ed

group f2

annali di

finite words

planar geometry

joint work

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English

Kassel Louis Pasteur University Institut de recherche mathématique avancée

From Christoffel words to braids

Christian Kassel

Institut de Recherche Mathématique Avancée CNRS  Université Louis Pasteur Strasbourg, France

Colloquium at Rutgers 13 April 2007

Joint work

•WithChristophe Reutenauer, UQAM, Montréal, Canada

•Paper published in Annali di Matematica Pura ed Applicata186 (2007), 317–339 Downloadable from arXiv:math.GR/0507219

What is it about?

Construct words from simple planar geometry ◮

◮ Words on two letters Christoffel words:ﬁnite words ◮ ◮ Sturmian sequences:inﬁnite words

In relation to ◮ bases of the free groupF2on two generators ◮ ◮ the braid groupB4on four strands

What is it about?

Construct words from simple planar geometry ◮

Words on two letters ◮ Christoffel words:ﬁnite words ◮ ◮ Sturmian sequences:inﬁnite words

◮ In relation to bases of the free groupF2on two generators ◮ the braid groupB4on four strands ◮

What is it about?

Construct words from simple planar geometry ◮

Words on two letters ◮ ◮ Christoffel words:ﬁnite words Sturmian sequences:inﬁnite words ◮

◮ In relation to ◮ bases of the free groupF2on two generators ◮ the braid groupB4on four strands

What is it about?

Construct words from simple planar geometry ◮

◮ Words on two letters ◮ Christoffel words:ﬁnite words ◮ Sturmian sequences:inﬁnite words

◮ In relation to bases of the free groupF2on two generators ◮ ◮ the braid groupB4on four strands

What is it about?

Construct words from simple planar geometry ◮

Words on two letters ◮ Christoffel words:ﬁnite words ◮ Sturmian sequences:inﬁnite words ◮

◮ In relation to ◮ bases of the free groupF2on two generators ◮ the braid groupB4on four strands

What is it about?

◮ Construct words from simple planar geometry

Words on two letters ◮ Christoffel words:ﬁnite words ◮ ◮ Sturmian sequences:inﬁnite words

◮ In relation to ◮ bases of the free groupF2on two generators the braid groupB4on four strands ◮

What is it about?

◮ Construct words from simple planar geometry

◮ Words on two letters ◮ Christoffel words:ﬁnite words ◮ Sturmian sequences:inﬁnite words

◮ In relation to ◮ bases of the free groupF2on two generators ◮ the braid groupB4on four strands

I. HISTORICAL BACKGROUND