<s>
In	O
mathematics	O
,	O
a	O
matrix	B-Algorithm
group	I-Algorithm
is	O
a	O
group	O
G	O
consisting	O
of	O
invertible	O
matrices	B-Architecture
over	O
a	O
specified	O
field	O
K	O
,	O
with	O
the	O
operation	O
of	O
matrix	O
multiplication	O
.	O
</s>
<s>
A	O
linear	B-Algorithm
group	I-Algorithm
is	O
a	O
group	O
that	O
is	O
isomorphic	O
to	O
a	O
matrix	B-Algorithm
group	I-Algorithm
(	O
that	O
is	O
,	O
admitting	O
a	O
faithful	O
,	O
finite-dimensional	O
representation	O
over	O
K	O
)	O
.	O
</s>
<s>
Any	O
finite	O
group	O
is	O
linear	O
,	O
because	O
it	O
can	O
be	O
realized	O
by	O
permutation	B-Algorithm
matrices	I-Algorithm
using	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
.	O
</s>
<s>
Among	O
infinite	O
groups	O
,	O
linear	B-Algorithm
groups	I-Algorithm
form	O
an	O
interesting	O
and	O
tractable	O
class	O
.	O
</s>
<s>
A	O
group	O
G	O
is	O
said	O
to	O
be	O
linear	O
if	O
there	O
exists	O
a	O
field	O
K	O
,	O
an	O
integer	O
d	O
and	O
an	O
injective	O
homomorphism	O
from	O
G	O
to	O
the	O
general	O
linear	B-Algorithm
group	I-Algorithm
GLd(K )	O
(	O
a	O
faithful	O
linear	O
representation	O
of	O
dimension	O
d	O
over	O
K	O
)	O
:	O
if	O
needed	O
one	O
can	O
mention	O
the	O
field	O
and	O
dimension	O
by	O
saying	O
that	O
G	O
is	O
linear	O
of	O
degree	O
d	O
over	O
K	O
.	O
Basic	O
instances	O
are	O
groups	O
which	O
are	O
defined	O
as	O
subgroups	O
of	O
a	O
linear	B-Algorithm
group	I-Algorithm
,	O
for	O
example	O
:	O
</s>
<s>
The	O
special	O
linear	B-Algorithm
group	I-Algorithm
SLn(K )	O
(	O
the	O
subgroup	O
of	O
matrices	B-Architecture
with	O
determinant	O
1	O
)	O
;	O
</s>
<s>
If	O
gi	O
is	O
a	O
collection	O
of	O
elements	O
in	O
GLn(K )	O
indexed	O
by	O
a	O
set	O
I	O
,	O
then	O
the	O
subgroup	O
generated	O
by	O
the	O
gi	O
is	O
a	O
linear	B-Algorithm
group	I-Algorithm
.	O
</s>
<s>
For	O
example	O
,	O
the	O
group	O
PSL2(R )	O
is	O
not	O
a	O
group	O
of	O
2	O
×	O
2	O
matrices	B-Architecture
,	O
but	O
it	O
has	O
a	O
faithful	O
representation	O
as	O
3	O
×	O
3	O
matrices	B-Architecture
(	O
the	O
adjoint	O
representation	O
)	O
,	O
which	O
can	O
be	O
used	O
in	O
the	O
general	O
case	O
.	O
</s>
<s>
Discrete	O
subgroups	O
of	O
classical	O
Lie	O
groups	O
(	O
for	O
example	O
lattices	O
or	O
thin	O
groups	O
)	O
are	O
also	O
examples	O
of	O
interesting	O
linear	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
A	O
finite	O
group	O
G	O
of	O
order	O
n	O
is	O
linear	O
of	O
degree	O
at	O
most	O
n	O
over	O
any	O
field	O
K	O
.	O
This	O
statement	O
is	O
sometimes	O
called	O
Cayley	B-Algorithm
's	I-Algorithm
theorem	I-Algorithm
,	O
and	O
simply	O
results	O
from	O
the	O
fact	O
that	O
the	O
action	O
of	O
G	O
on	O
the	O
group	O
ring	O
K[G]	O
by	O
left	O
(	O
or	O
right	O
)	O
multiplication	O
is	O
linear	O
and	O
faithful	O
.	O
</s>
<s>
While	O
example	O
4	O
above	O
is	O
too	O
general	O
to	O
define	O
a	O
distinctive	O
class	O
(	O
it	O
includes	O
all	O
linear	B-Algorithm
groups	I-Algorithm
)	O
,	O
restricting	O
to	O
a	O
finite	O
index	O
set	O
I	O
,	O
that	O
is	O
,	O
to	O
finitely	O
generated	O
groups	O
allows	O
to	O
construct	O
many	O
interesting	O
examples	O
.	O
</s>
<s>
The	O
ping-pong	O
lemma	O
can	O
be	O
used	O
to	O
construct	O
many	O
examples	O
of	O
linear	B-Algorithm
groups	I-Algorithm
which	O
are	O
free	O
groups	O
(	O
for	O
instance	O
the	O
group	O
generated	O
by	O
is	O
free	O
)	O
.	O
</s>
<s>
Braid	B-Application
groups	I-Application
(	O
which	O
are	O
defined	O
as	O
a	O
finitely	O
presented	O
group	O
)	O
have	O
faithful	O
linear	O
representation	O
on	O
a	O
finite-dimensional	O
complex	O
vector	O
space	O
where	O
the	O
generators	O
act	O
by	O
explicit	O
matrices	B-Architecture
.	O
</s>
<s>
In	O
some	O
cases	O
the	O
fundamental	O
group	O
of	O
a	O
manifold	B-Architecture
can	O
be	O
shown	O
to	O
be	O
linear	O
by	O
using	O
representations	O
coming	O
from	O
a	O
geometric	O
structure	O
.	O
</s>
<s>
A	O
generalization	O
of	O
this	O
construction	O
is	O
given	O
by	O
the	O
notion	O
of	O
a	O
(	O
G	O
,	O
X	O
)	O
-structure	O
on	O
a	O
manifold	B-Architecture
.	O
</s>
<s>
Another	O
example	O
is	O
the	O
fundamental	O
group	O
of	O
Seifert	O
manifolds	B-Architecture
.	O
</s>
<s>
While	O
linear	B-Algorithm
groups	I-Algorithm
are	O
a	O
vast	O
class	O
of	O
examples	O
,	O
among	O
all	O
infinite	O
groups	O
they	O
are	O
distinguished	O
by	O
many	O
remarkable	O
properties	O
.	O
</s>
<s>
Finitely	O
generated	O
linear	B-Algorithm
groups	I-Algorithm
have	O
the	O
following	O
properties	O
:	O
</s>
<s>
Schur	O
's	O
theorem	O
:	O
a	O
torsion	O
linear	B-Algorithm
group	I-Algorithm
is	O
locally	O
finite	O
.	O
</s>
<s>
Selberg	O
's	O
lemma	O
:	O
any	O
finitely	O
generated	O
linear	B-Algorithm
group	I-Algorithm
contains	O
a	O
torsion-free	O
subgroup	O
of	O
finite	O
index	O
.	O
</s>
<s>
The	O
Tits	O
alternative	O
states	O
that	O
a	O
linear	B-Algorithm
group	I-Algorithm
either	O
contains	O
a	O
non-abelian	O
free	O
group	O
or	O
else	O
is	O
virtually	O
solvable	O
(	O
that	O
is	O
,	O
contains	O
a	O
solvable	O
group	O
of	O
finite	O
index	O
)	O
.	O
</s>
<s>
the	O
Dehn	O
function	O
of	O
a	O
finitely	O
generated	O
linear	B-Algorithm
group	I-Algorithm
can	O
only	O
be	O
either	O
polynomial	O
or	O
exponential	O
;	O
</s>
<s>
an	O
amenable	O
linear	B-Algorithm
group	I-Algorithm
is	O
virtually	O
solvable	O
,	O
in	O
particular	O
elementary	O
amenable	O
;	O
</s>
<s>
the	O
von	O
Neumann	O
conjecture	O
is	O
true	O
for	O
linear	B-Algorithm
groups	I-Algorithm
.	O
</s>
<s>
Since	O
the	O
symmetric	B-Algorithm
group	I-Algorithm
on	O
an	O
infinite	O
set	O
contains	O
this	O
group	O
it	O
is	O
also	O
not	O
linear	O
.	O
</s>
<s>
Since	O
any	O
finitely	O
linear	B-Algorithm
group	I-Algorithm
is	O
residually	O
finite	O
,	O
it	O
cannot	O
be	O
both	O
simple	O
and	O
infinite	O
.	O
</s>
<s>
The	O
outer	B-Algorithm
automorphism	I-Algorithm
group	I-Algorithm
Out(Fn )	O
of	O
the	O
free	O
group	O
is	O
known	O
not	O
to	O
be	O
linear	O
for	O
n	O
at	O
least	O
4	O
.	O
</s>
<s>
In	O
contrast	O
with	O
the	O
case	O
of	O
braid	B-Application
groups	I-Application
,	O
it	O
is	O
an	O
open	O
question	O
whether	O
the	O
mapping	O
class	O
group	O
of	O
a	O
surface	O
of	O
genus	O
>	O
1	O
is	O
linear	O
.	O
</s>
<s>
The	O
representation	O
theory	O
of	O
infinite	O
finitely	O
generated	O
groups	O
is	O
in	O
general	O
mysterious	O
;	O
the	O
object	O
of	O
interest	O
in	O
this	O
case	O
are	O
the	O
character	B-Algorithm
varieties	I-Algorithm
of	O
the	O
group	O
,	O
which	O
are	O
well	O
understood	O
only	O
in	O
very	O
few	O
cases	O
,	O
for	O
example	O
free	O
groups	O
,	O
surface	O
groups	O
and	O
more	O
generally	O
lattices	O
in	O
Lie	O
groups	O
(	O
for	O
example	O
through	O
Margulis	O
 '	O
superrigidity	O
theorem	O
and	O
other	O
rigidity	O
results	O
)	O
.	O
</s>
