<s>
In	O
the	O
mathematical	O
area	O
of	O
group	O
theory	O
,	O
Artin	B-Algorithm
groups	I-Algorithm
,	O
also	O
known	O
as	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
or	O
generalized	O
braid	B-Application
groups	I-Application
,	O
are	O
a	O
family	O
of	O
infinite	O
discrete	O
groups	O
defined	O
by	O
simple	O
presentations	O
.	O
</s>
<s>
They	O
are	O
closely	O
related	O
with	O
Coxeter	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Examples	O
are	O
free	O
groups	O
,	O
free	O
abelian	O
groups	O
,	O
braid	B-Application
groups	I-Application
,	O
and	O
right-angled	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
,	O
among	O
others	O
.	O
</s>
<s>
The	O
groups	O
are	O
named	O
after	O
Emil	O
Artin	O
,	O
due	O
to	O
his	O
early	O
work	O
on	O
braid	B-Application
groups	I-Application
in	O
the	O
1920s	O
to	O
1940s	O
,	O
and	O
Jacques	O
Tits	O
who	O
developed	O
the	O
theory	O
of	O
a	O
more	O
general	O
class	O
of	O
groups	O
in	O
the	O
1960s	O
.	O
</s>
<s>
An	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
is	O
a	O
group	O
that	O
admits	O
an	O
Artin	O
–	O
Tits	O
presentation	O
.	O
</s>
<s>
Alternatively	O
,	O
an	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
can	O
be	O
specified	O
by	O
the	O
set	O
of	O
generators	O
and	O
,	O
for	O
every	O
in	O
,	O
the	O
natural	O
number	O
that	O
is	O
the	O
length	O
of	O
the	O
words	O
and	O
such	O
that	O
is	O
the	O
relation	O
connecting	O
and	O
,	O
if	O
any	O
.	O
</s>
<s>
The	O
integers	O
can	O
be	O
organized	O
into	O
a	O
symmetric	B-Algorithm
matrix	I-Algorithm
,	O
known	O
as	O
the	O
Coxeter	O
matrix	O
of	O
the	O
group	O
.	O
</s>
<s>
If	O
is	O
an	O
Artin	O
–	O
Tits	O
presentation	O
of	O
an	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
,	O
the	O
quotient	O
of	O
obtained	O
by	O
adding	O
the	O
relation	O
for	O
each	O
of	O
is	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
Conversely	O
,	O
if	O
is	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
presented	O
by	O
reflections	O
and	O
the	O
relations	O
are	O
removed	O
,	O
the	O
extension	O
thus	O
obtained	O
is	O
an	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
.	O
</s>
<s>
For	O
instance	O
,	O
the	O
Coxeter	B-Algorithm
group	I-Algorithm
associated	O
with	O
the	O
-strand	O
braid	B-Application
group	I-Application
is	O
the	O
symmetric	B-Algorithm
group	O
of	O
all	O
permutations	O
of	O
.	O
</s>
<s>
is	O
the	O
braid	B-Application
group	I-Application
on	O
strands	O
;	O
here	O
for	O
,	O
and	O
for	O
.	O
</s>
<s>
If	O
is	O
an	O
Artin	O
–	O
Tits	O
monoid	O
,	O
and	O
if	O
is	O
the	O
associated	O
Coxeter	B-Algorithm
group	I-Algorithm
,	O
there	O
is	O
a	O
(	O
set-theoretic	O
)	O
section	O
of	O
into	O
,	O
and	O
every	O
element	O
of	O
admits	O
a	O
distinguished	O
decomposition	O
as	O
a	O
sequence	O
of	O
elements	O
in	O
the	O
image	O
of	O
(	O
"	O
greedy	O
normal	O
form	O
"	O
)	O
.	O
</s>
<s>
Very	O
few	O
results	O
are	O
known	O
for	O
general	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
are	O
infinite	O
countable	O
.	O
</s>
<s>
In	O
an	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
,	O
the	O
only	O
relation	O
connecting	O
the	O
squares	O
of	O
the	O
elements	O
of	O
is	O
if	O
is	O
in	O
(	O
John	O
Crisp	O
and	O
Luis	O
Paris	O
)	O
.	O
</s>
<s>
For	O
every	O
Artin	O
–	O
Tits	O
presentation	O
,	O
the	O
Artin	O
–	O
Tits	O
monoid	O
presented	O
by	O
embeds	O
in	O
the	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
presented	O
by	O
(	O
Paris	O
)	O
.	O
</s>
<s>
Several	O
important	O
classes	O
of	O
Artin	B-Algorithm
groups	I-Algorithm
can	O
be	O
defined	O
in	O
terms	O
of	O
the	O
properties	O
of	O
the	O
Coxeter	O
matrix	O
.	O
</s>
<s>
An	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
is	O
said	O
to	O
be	O
of	O
spherical	O
type	O
if	O
the	O
associated	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
finite	O
—	O
the	O
alternative	O
terminology	O
"	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
of	O
finite	O
type	O
"	O
is	O
to	O
be	O
avoided	O
,	O
because	O
of	O
its	O
ambiguity	O
:	O
a	O
"	O
finite	O
type	O
group	O
"	O
is	O
just	O
one	O
that	O
admits	O
a	O
finite	O
generating	O
set	O
.	O
</s>
<s>
In	O
the	O
case	O
of	O
a	O
spherical	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
,	O
the	O
group	O
is	O
a	O
group	O
of	O
fractions	O
for	O
the	O
monoid	O
,	O
making	O
the	O
study	O
much	O
easier	O
.	O
</s>
<s>
Every	O
above-mentioned	O
problem	O
is	O
solved	O
in	O
the	O
positive	O
for	O
spherical	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
:	O
the	O
word	O
and	O
conjugacy	O
problems	O
are	O
decidable	O
,	O
their	O
torsion	O
is	O
trivial	O
,	O
the	O
center	O
is	O
monogenic	O
in	O
the	O
irreducible	O
case	O
,	O
and	O
the	O
cohomology	O
is	O
determined	O
(	O
Pierre	O
Deligne	O
,	O
by	O
geometrical	O
methods	O
,	O
Egbert	O
Brieskorn	O
and	O
Kyoji	O
Saito	O
,	O
by	O
combinatorial	O
methods	O
)	O
.	O
</s>
<s>
A	O
pure	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
of	O
spherical	O
type	O
can	O
be	O
realized	O
as	O
the	O
fundamental	O
group	O
of	O
the	O
complement	O
of	O
a	O
finite	O
hyperplane	O
arrangement	O
in	O
.	O
</s>
<s>
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
of	O
spherical	O
type	O
are	O
biautomatic	O
groups	O
(	O
Ruth	O
Charney	O
)	O
.	O
</s>
<s>
An	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
is	O
said	O
to	O
be	O
right-angled	O
if	O
all	O
coefficients	O
of	O
the	O
Coxeter	O
matrix	O
are	O
either	O
or	O
,	O
i.e.	O
,	O
all	O
relations	O
are	O
commutation	O
relations	O
.	O
</s>
<s>
The	O
names	O
(	O
free	O
)	O
partially	O
commutative	O
group	O
,	O
graph	B-Algorithm
group	I-Algorithm
,	O
trace	B-Algorithm
group	I-Algorithm
,	O
semifree	B-Algorithm
group	I-Algorithm
or	O
even	O
locally	B-Algorithm
free	I-Algorithm
group	I-Algorithm
are	O
also	O
common	O
.	O
</s>
<s>
For	O
this	O
class	O
of	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
,	O
a	O
different	O
labeling	O
scheme	O
is	O
commonly	O
used	O
.	O
</s>
<s>
The	O
class	O
of	O
right-angled	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
includes	O
the	O
free	O
groups	O
of	O
finite	O
rank	O
,	O
corresponding	O
to	O
a	O
graph	O
with	O
no	O
edges	O
,	O
and	O
the	O
finitely-generated	O
free	O
abelian	O
groups	O
,	O
corresponding	O
to	O
a	O
complete	O
graph	O
.	O
</s>
<s>
Every	O
right-angled	O
Artin	B-Algorithm
group	I-Algorithm
of	O
rank	O
r	O
can	O
be	O
constructed	O
as	O
HNN	O
extension	O
of	O
a	O
right-angled	O
Artin	B-Algorithm
group	I-Algorithm
of	O
rank	O
,	O
with	O
the	O
free	O
product	O
and	O
direct	O
product	O
as	O
the	O
extreme	O
cases	O
.	O
</s>
<s>
A	O
right-angled	O
Artin	B-Algorithm
group	I-Algorithm
is	O
a	O
special	O
case	O
of	O
this	O
product	O
,	O
with	O
every	O
vertex/operand	O
of	O
the	O
graph-product	O
being	O
a	O
free	O
group	O
of	O
rank	O
one	O
(	O
the	O
infinite	O
cyclic	O
group	O
)	O
.	O
</s>
<s>
The	O
word	O
and	O
conjugacy	O
problems	O
of	O
a	O
right-angled	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
are	O
decidable	O
,	O
the	O
former	O
in	O
linear	O
time	O
,	O
the	O
group	O
is	O
torsion-free	O
,	O
and	O
there	O
is	O
an	O
explicit	O
cellular	O
finite	O
(	O
John	O
Crisp	O
,	O
Eddy	O
Godelle	O
,	O
and	O
Bert	O
Wiest	O
)	O
.	O
</s>
<s>
Every	O
right-angled	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
acts	O
freely	O
and	O
cocompactly	O
on	O
a	O
finite-dimensional	O
CAT(0 )	O
cube	O
complex	O
,	O
its	O
"	O
Salvetti	O
complex	O
"	O
.	O
</s>
<s>
As	O
an	O
application	O
,	O
one	O
can	O
use	O
right-angled	O
Artin	B-Algorithm
groups	I-Algorithm
and	O
their	O
Salvetti	O
complexes	O
to	O
construct	O
groups	O
with	O
given	O
finiteness	O
properties	O
(	O
Mladen	O
Bestvina	O
and	O
Noel	O
Brady	O
)	O
see	O
also	O
(	O
Ian	O
Leary	O
)	O
.	O
</s>
<s>
An	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
(	O
and	O
a	O
Coxeter	B-Algorithm
group	I-Algorithm
)	O
is	O
said	O
to	O
be	O
of	O
large	O
type	O
if	O
for	O
all	O
generators	O
;	O
it	O
is	O
said	O
to	O
be	O
of	O
extra-large	O
type	O
if	O
for	O
all	O
generators	O
.	O
</s>
<s>
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
of	O
extra-large	O
type	O
are	O
eligible	O
for	O
small	O
cancellation	O
theory	O
.	O
</s>
<s>
As	O
an	O
application	O
,	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
of	O
extra-large	O
type	O
are	O
torsion-free	O
and	O
have	O
solvable	O
conjugacy	O
problem	O
(	O
Kenneth	O
Appel	O
and	O
Paul	O
Schupp	O
)	O
.	O
</s>
<s>
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
of	O
extra-large	O
type	O
are	O
biautomatic	O
(	O
David	O
Peifer	O
)	O
.	O
</s>
<s>
Artin	B-Algorithm
groups	I-Algorithm
of	O
large	O
type	O
are	O
shortlex	O
automatic	O
with	O
regular	O
geodesics	O
(	O
Derek	O
Holt	O
and	O
Sarah	O
Rees	O
)	O
.	O
</s>
<s>
Many	O
other	O
families	O
of	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
have	O
been	O
identified	O
and	O
investigated	O
.	O
</s>
<s>
An	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
is	O
said	O
to	O
be	O
of	O
FC	O
type	O
(	O
"	O
flag	O
complex	O
"	O
)	O
if	O
,	O
for	O
every	O
subset	O
of	O
such	O
that	O
for	O
all	O
in	O
,	O
the	O
group	O
is	O
of	O
spherical	O
type	O
.	O
</s>
<s>
An	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
group	I-Algorithm
is	O
said	O
to	O
be	O
of	O
affine	O
type	O
if	O
the	O
associated	O
Coxeter	B-Algorithm
group	I-Algorithm
is	O
affine	O
.	O
</s>
<s>
Affine	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
are	O
of	O
Euclidean	O
type	O
:	O
the	O
associated	O
Coxeter	B-Algorithm
group	I-Algorithm
acts	O
geometrically	O
on	O
a	O
Euclidean	O
space	O
.	O
</s>
<s>
In	O
2019	O
,	O
a	O
proof	O
of	O
the	O
conjecture	O
was	O
announced	O
for	O
all	O
affine	O
Artin	B-Algorithm
–	I-Algorithm
Tits	I-Algorithm
groups	I-Algorithm
(	O
Mario	O
Salvetti	O
and	O
Giovanni	O
Paolini	O
)	O
.	O
</s>
