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- Coxeter Groups are Biautomatic: Lecture 2 by Piotr Przytycki & Damian Osajda
Coxeter Groups are Biautomatic: Lecture 2 by Piotr Przytycki & Damian Osajda
Learn about the concept of Coxeter groups being biautomatic in this informative lecture by Piotr Przytycki and Damian Osajda. Discover the criteria for a group to be considered biautomatic and the properties it entails, such as having a generating set and regular language. Dive into the details of how every group element has a representative in the language and the fascinating fellow traveler property.
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Video Summary & Chapters
Video Transcript
first talk of today is uh P's second
talk on Cox to groups being biomatic
thank
you so first a small recap from last
time a group is by automatic if there is
a generating set s and the regular
language over s such that the map from
the language to the group is subjective
every word has a represent every group
element has a representative in this
language and fellow traveler property
holds so for every pair of words in the
language that are equal in the group or
the fair by a generator on the right
these words feral travel and second part
if they defer on the left in the group
then the words W and s w Prime below
travel
and we Pro the corollary last time which
is true even for automatic groups so
just for groups satisfying part one that
one implies
that any word any group
element in
the uh automatic group represented by a
word that might not necessarily be in
the language can be still represented
efficiently can be represented by a word
in the language of length at most some
constant times the length of the
original word plus and
zero and today we are supposed to prove
the theorem that every automatic group
so satisfying just part
one is finally presented we assume it's
finally generated but for the moment we
don't know it's Finly presented and
moreover it satisfies quadratic
isoparametric inequality which means
that there is a constant C such that for
any word representing the trial element
in the group this word can be
represented efficiently as a product of
conjugates of
relators in the free group namely the
number of these conjugates it say most
quadratic in the length of the
word any questions before we start the
proof okay so let's do the
proof so we have a
Video Summary & Chapters
Video Transcript
first talk of today is uh P's second
talk on Cox to groups being biomatic
thank
you so first a small recap from last
time a group is by automatic if there is
a generating set s and the regular
language over s such that the map from
the language to the group is subjective
every word has a represent every group
element has a representative in this
language and fellow traveler property
holds so for every pair of words in the
language that are equal in the group or
the fair by a generator on the right
these words feral travel and second part
if they defer on the left in the group
then the words W and s w Prime below
travel
and we Pro the corollary last time which
is true even for automatic groups so
just for groups satisfying part one that
one implies
that any word any group
element in
the uh automatic group represented by a
word that might not necessarily be in
the language can be still represented
efficiently can be represented by a word
in the language of length at most some
constant times the length of the
original word plus and
zero and today we are supposed to prove
the theorem that every automatic group
so satisfying just part
one is finally presented we assume it's
finally generated but for the moment we
don't know it's Finly presented and
moreover it satisfies quadratic
isoparametric inequality which means
that there is a constant C such that for
any word representing the trial element
in the group this word can be
represented efficiently as a product of
conjugates of
relators in the free group namely the
number of these conjugates it say most
quadratic in the length of the
word any questions before we start the
proof okay so let's do the
proof so we have a
