ombinatorial roups - ibm · ombinatorial a spects of f inite s imple g roups shelly garion ......
TRANSCRIPT
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
CO
MP
UT
AT
ION
AL
AN
D C
OM
BIN
AT
OR
IAL
AS
PE
CT
S O
F F
INIT
E S
IMP
LE
GR
OU
PS
Sh
elly
Ga
rio
n
SF
B 8
78
– G
rou
ps, G
eo
metr
y &
Actio
ns
Univ
ers
ität
Münste
r
http
://w
ww
math
.un
i-m
ue
nste
r.d
e/u
/sh
elly
.ga
rio
n/
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
Fin
ite
S
imp
le
Gro
up
s
Gro
up
Th
eo
ry��
Alg
eb
raic
G
eo
me
try
&��
Nu
mb
er
Th
eory
��
Re
pre
se
nta
tio
n
Th
eory
��
&
��
Ch
ara
cte
rs��
Dyn
am
ics
&��
Actio
ns
��
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
Fin
ite
sim
ple
gro
ups
Th
e fin
ite
sim
ple
gro
ups a
re th
e b
uild
ing b
locks o
f a
ll fin
ite
gro
up
s.
De
fin
itio
n. G
is s
imp
le if it h
as n
o n
on
-tri
via
l n
orm
al sub
gro
ups.
Th
eore
m. C
lassific
atio
n o
f th
e (
non
-ab
elia
n)
Fin
ite
Sim
ple
Gro
ups.
� A
lte
rna
tin
g g
rou
ps A
n (
n�5
).
� F
inite
sim
ple
gro
up
s o
f L
ie typ
e G
r(q
)
wh
ere
r is th
e L
ie r
ank a
nd
q=
pe is th
e s
ize
of fin
ite
fie
ld; e
.g. P
SL
r+1(q
).
� 2
6 s
po
rad
ic g
rou
ps.
On
th
e p
roo
f… th
ou
sa
nds o
f p
ag
es, h
un
dre
ds o
f art
icle
s, �1
00
auth
ors
:
Fro
m G
alo
is (
18
32)
to G
ore
nste
in-L
yons-S
olo
mo
n (
90
's)…
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
Th
e P
rod
uct
Re
pla
ce
me
nt
Alg
ori
thm
The p
roble
m
Ba
sic
pro
ble
m in
co
mpu
tatio
na
l g
rou
p th
eo
ry:
How
to
ge
nera
te a
ra
ndo
m e
lem
en
t in
a fin
ite
gro
up
G?
Th
e
Pro
du
ct
Re
pla
ce
men
t A
lgo
rith
m
(PR
A)
wa
s
su
gge
ste
d
in
19
95
by
Ce
ller,
Le
edha
m-G
reen
, M
urr
ay, N
iem
eye
r &
O'B
rie
n.
Th
e P
RA
show
ed
ve
ry g
ood
pe
rform
ance
in
pra
ctica
l e
xp
eri
men
ts,
bu
t th
ere
is
no
rig
oro
us justifica
tio
n. It w
as in
clu
de
d in
GA
P a
nd
MA
GM
A.
Th
e P
RA
pe
rfo
rms a
ran
do
m w
alk
on
th
e p
rod
uct
rep
lace
men
t g
raph
�n(G
) w
hose
ve
rtic
es a
re th
e g
ene
ratin
g n
-tu
ple
s o
f G
(fo
r a
fix
ed
n):
{ (g
1,…
,gn)
: <
g1,…
,gn>
=G
}
Qu
estio
n: L
et G
be
a fin
ite
sim
ple
gro
up
. Is
�n(G
) co
nn
ecte
d?
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
Th
e P
rod
uct
Re
pla
ce
me
nt
Alg
ori
thm
The r
esults
n=
2
n�3
[Ne
um
an
n, 19
51
].
�2(A
5)
is d
isco
nn
ecte
d.
[Ga
rio
n-S
ha
lev, 2
00
9].
(Con
jectu
red
in
20
02
by
[Gu
raln
ick-P
ak] )
.
If G
is a
fin
ite
sim
ple
gro
up
,
the
n th
e n
um
be
r o
f con
necte
d
co
mpo
nen
ts o
f �
2(G
) gro
ws to
infin
ity a
s |G
|��
.
Wie
go
ld's
con
jectu
re (
19
80
's).
If G
is a
fin
ite
sim
ple
gro
up
an
d n�3
th
en
�n(G
) is
co
nne
cte
d.
�n(G
) is
co
nne
cte
d:
� [G
ilma
n, 1
977
]. G
=P
SL
2(p
), n�3
.
� [E
va
ns, 1
99
3]. G
=P
SL
2(2
e),
Sz(2
2e+
1),
n�3
.
� [G
ari
on
, 2
00
8]. G
=P
SL
2(p
e),
n�4
.
� [A
vn
i-G
ari
on
, 2
00
8]. G
=G
r(p
e),
n�c
(r),
fin
ite
sim
ple
gro
up
of L
ie typ
e o
f L
ie r
ank r
.
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
Th
e P
rod
uct
Re
pla
ce
me
nt
Alg
ori
thm
The m
eth
ods
Is �
n(G
) con
ne
cte
d?
�
Ca
n w
e c
on
ne
ct a
ny g
en
era
tin
g n
-tu
ple
of G
to
a r
ed
un
da
nt o
ne?
�
Wh
at a
re th
e s
ubg
rou
ps o
f G
?
[Ga
rio
n, 2
00
8]. F
or
G=
PS
L2(q
): th
e s
ub
gro
ups a
re w
ell-
kno
wn
[D
ickso
n, 1
90
1].
[Avn
i-G
ari
on
, 2
00
8]. F
or
G=
Gr(q
) –
a fin
ite
sim
ple
gro
up
of L
ie typ
e:
� A
schb
ach
er's c
lassific
atio
n o
f m
axim
al su
bg
rou
ps (
198
4)
– u
ses C
FS
G.
� [L
ars
en-P
ink, 1
99
8] –
use
s a
lge
bra
ic g
eo
me
try (
no
t C
FS
G!)
.
[Ga
rio
n-G
lasn
er]
. F
or
an
in
fin
ite
sim
ple
Ta
rski m
on
ste
r g
rou
p G
:
� A
ny s
ub
gro
up
is a
cyclic
gro
up
of ord
er
p (
p is e
ith
er
a fix
ed
pri
me
or �
).
� T
he
ore
m. A
fa
ith
ful h
igh
ly tra
nsitiv
e a
ction
of O
ut(
Fn)
on
a c
oun
tab
le s
et.
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
Wo
rd m
ap
s in
gro
up
s
The p
roble
m
A w
ord
w=
w(x
1,…
,xn)
is a
n e
lem
en
t in
Fn=
<x
1,…
,xn>
.
Fo
r a
gro
up
G, a
wo
rd m
ap
is g
ive
n b
y:
w: G
n �
G
(g1,…
,gn) �
w(g
1,…
,gn)
Qu
estio
ns: L
et G
be
a fin
ite
sim
ple
gro
up
an
d le
t w�1
.
� W
ha
t is
th
e im
ag
e w
(Gn)?
Is w
: Gn �
G s
urj
ective
?
� W
ha
t a
re th
e s
ize
s o
f th
e fib
ers
of a
word
ma
p?
Is w
: Gn�
G a
lmo
st e
qu
idis
trib
ute
d?
e.g
. |w
-1(g
)|
|G
|n-1
fo
r a
lmo
st a
ll g
G.
Inve
stig
ate
d b
y G
ura
lnic
k, L
ars
en
, L
ieb
eck, S
eg
al, S
ha
lev, O
'Bri
en
, T
iep
, …
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
Wo
rd m
ap
s in
gro
up
s
The r
esults
Co
mm
uta
tor
wo
rd: e
1 =
[x,y
] =
xyx
-1y
-1
F2
Th
eore
m (
Ore
's C
on
jectu
re)
[Ore
, 1
951
; T
ho
mpso
n,
19
60
's;
Elle
rs-G
ord
ee
v,
19
98
; L
iebe
ck-O
'Bri
en
-Sha
lev-T
iep
, 2
00
8; …
]
An
y e
lem
en
t in
a fin
ite
sim
ple
gro
up
is a
co
mm
uta
tor.
[Ga
rio
n-S
ha
lev, 2
00
9].
Th
e c
om
mu
tato
r m
ap
on
fin
ite
sim
ple
gro
up
s is a
lmost e
qu
idis
trib
ute
d.
En
ge
l w
ord
s: e
n =
[e
n-1
,y]
= […
[[x,y
],y],…
,y]
F2
[Ba
nd
ma
n-G
ari
on-G
rune
wa
ld]. S
urj
ectivity a
nd
eq
uid
istr
ibu
tion
of e
n o
n P
SL
2(q
).
Tw
o-p
ow
er
wo
rds: w
=x
ay
b
F2 [G
ura
lnic
k-M
alle
, 2
012
; L
OS
T, 2
012
].
[Ba
nd
ma
n-G
ari
on
, 2
01
2]. S
urj
ectivity a
nd
eq
uid
istr
ibu
tio
n o
f x
ay
b o
n P
SL
2(q
).
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
Wo
rd m
ap
s in
gro
up
s
The m
eth
ods
Co
mm
uta
tor
wo
rd:
[Fro
ben
ius, 1
89
6]. #
{(x
,y)
G�G
: [x
,y]=
g}
= |G
| �
�
Irr(
G) �(
g)/�(
1)
Wo
rds in
SL
2(q
):
Tra
ce
ma
p T
he
ore
m [F
ricke
-Kle
in, 1
897; V
og
t, 1
88
9].
wo
rd w
(x,y
) in
SL
2(q
) �
tr
(w)=
P(s
,t,u
) is
a p
oly
no
mia
l in
s=
tr(x
), t=
tr(y
), u
=tr
(xy)
over
Fq
Exa
mp
les:
w
=[x
,y]
�
tr(w
) =
s2 +
t2 +
u2 -
stu
- 2
[Ba
nd
ma
n-G
ari
on-G
rune
wa
ld]. w
=e
n(x
,y) �
s
n =
tr(
en)
= s
n-1
2 +
2t2
- s
n-1
t2 -
2
[Ba
nd
ma
n-G
ari
on
, 2
01
2].
w=
xay
b
By in
du
ctio
n, co
mpu
te tr(
w)
for
an
y w
F2
�
tr(w
) =
u�f a
,b(s
,t)
+ h
a,b(s
,t)
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
�
Be
au
vill
e s
urf
ace
s
The p
roble
m
Be
au
vill
e s
urf
ace
[B
ea
uvill
e, 1
97
8; C
ata
ne
se
, 2
00
0]. S
=(C
1�C
2)/
G
S i
s a
n i
nfin
ite
sim
ally
rig
id c
om
ple
x s
urf
ace
, w
here
C1 a
nd
C2 a
re c
urv
es o
f
ge
nus �
2 a
nd G
is a
fin
ite
gro
up
actin
g fre
ely
on
th
eir
pro
du
ct.
Be
au
vill
e s
tructu
re [B
aue
r-C
ata
ne
se-G
run
ew
ald
, 2
005
]. (
x1,y
1,z
1;x
2,y
2,z
2)
� x
1y
1z
1 =
1 =
x2y
2z
2 ,
� <
x1,y
1>
= G
= <
x2,y
2>
,
� n
o n
on-i
den
tity
po
we
r o
f x
1,y
1,z
1 is c
on
jug
ate
in
G to
a p
ow
er
of x
2,y
2,z
2.
Th
e typ
e o
f (x
1,y
1,z
1;x
2,y
2,z
2)
is th
e 6
ord
ers
of x
1,y
1,z
1;x
2,y
2,z
2.
Qu
estio
ns [B
au
er-
Ca
tan
ese
-Gru
ne
wa
ld, 2
00
5].
1)
Wh
ich
fin
ite
sim
ple
gro
up
s a
dm
it a
Bea
uvill
e s
tructu
re?
2)
Wh
ich
typ
es c
an
occur
in a
Be
au
vill
e s
tructu
re?
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
��
Be
au
vill
e s
urf
ace
s
The r
esults
Co
nje
ctu
re 1
[B
CG
, 2
00
5].
All fin
ite
sim
ple
gro
up
s (
exce
pt A
5)
ad
mit a
Be
au
vill
e s
tructu
re.
Pro
ve
d fo
r:
� A
n (
n�6
) [B
CG
, 2
00
5; F
uert
es,G
on
zá
les-D
iez, 2
00
9].
� P
SL
2(q
) (q�7
) [F
ue
rte
s-J
one
s, 2
011
; G
ario
n-P
ene
gin
i].
� A
lmo
st
all fin
ite
sim
ple
gro
up
s [G
ari
on-L
ars
en-L
ubo
tzky, 2
01
2].
� A
ll fin
ite
sim
ple
gro
up
s (�A
5)
[Fa
irb
airn
-Ma
ga
ard
-Pa
rke
r; G
ura
lnic
k-M
alle
].
Co
nje
ctu
re 2
[B
CG
, 2
00
5] –
pro
ve
d b
y [G
ari
on
-Pen
eg
ini].
Fo
r a
ny
two
h
yp
erb
olic
tr
iple
s
of
inte
ge
rs
(k1,l
1,m
1;k
2,l
2,m
2)
alm
ost
all
alte
rna
ting
gro
ups A
n a
dm
it a
Be
au
vill
e s
tructu
re o
f ty
pe
(k
1,l
1,m
1;k
2,l
2,m
2).
[Ga
rio
n]. C
hara
cte
riza
tio
n o
f th
e typ
es o
f B
ea
uvill
e s
tru
ctu
res for
PS
L2(q
).
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
��
Be
au
vill
e s
urf
ace
s
The m
eth
ods
G a
dm
its a
Be
au
vill
e s
tru
ctu
re o
f ty
pe
(k
1,l
1,m
1;k
2,l
2,m
2).
�
G is a
qu
otien
t o
f �
(k1,l
1,m
1)
an
d �
(k2,l
2,m
2)
+ "
dis
join
t" c
on
ditio
n.
Tri
ang
le g
roup
: �
(k,l,m
) =
< x
,y:
xk =
yl =
(xy)m
=1
>
Qu
estio
n: W
hic
h fin
ite
sim
ple
gro
up
s a
re q
uo
tie
nts
of a
giv
en
�(k
,l,m
)?
� A
n –
Hig
ma
n 1
96
0s; C
onde
r 1
980
; E
ve
ritt 2
00
0; L
ieb
eck-S
ha
lev 2
00
4,…
� P
SL
2(q
) –
Macb
ea
th 1
96
8; R
ose
nbe
rge
r e
t a
l. 1
98
9; M
ari
on
20
09
.
� G
r(q)
– o
pe
n! L
uccin
i-T
am
bu
rin
i-W
ilso
n 2
00
0; L
ieb
eck-S
ha
lev 2
00
5,…
[Fro
ben
ius, 1
89
0s]. X
,Y,Z
– c
on
jug
acy c
lasses in
G (
of o
rde
rs k
,l,m
).
# {
x,y
,z: x
X,
y
Y, z
Z, xyz=
1}
= |X
|�|Y
|�|Z
|/|G
| �
�
Irr(
G) �(
x)�
(y)�
(z)/�(
1)
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
��
Fin
ite
S
imp
le
Gro
up
s
Gro
up
Th
eo
ry��
Alg
eb
raic
G
eo
me
try
&��
Nu
mb
er
Th
eory
��
Re
pre
se
nta
tio
n
Th
eory
��
&
��
Ch
ara
cte
rs��
Dyn
am
ics
&��
Actio
ns
��
����������
������
���������
����������������
��������
��������
����������������
�������
�����
������
����
�
�������
������������
������ ��
!����������"�
#�$#��
GA
P
Com
puta
tio
nal and c
om
bin
ato
rial aspects
of
finite s
imple
gro
ups
��
Fu
ture
pla
ns
Co
ntin
ue
my r
ese
arc
h i
n g
rou
p t
he
ory
, fo
cusin
g o
n f
inite
sim
ple
gro
ups,
wh
ile
inte
ractin
g
with
o
the
r fie
lds
of
ma
them
atics
su
ch
as
alg
ebra
ic
geo
me
try,
nu
mb
er
the
ory
, re
pre
se
nta
tio
n th
eo
ry, d
yn
am
ics…
So
me
spe
cific
rese
arc
h p
rob
lem
s…
Wo
rd m
aps
� A
na
lysis
of ge
ne
ral w
ord
s in
PS
L2(q
).
� G
en
era
lize
the
tra
ce
ma
p m
eth
od
to
PS
Ln(q
) an
d G
r(q
).
� G
en
era
liza
tion
s to
SL
2(Z
p)
an
d S
L2(Z
).
Be
au
vill
e s
urf
aces
� W
ha
t is
th
e p
rob
ab
ility
of a
dm
ittin
g a
Be
au
vill
e s
tructu
re?
� C
onstr
uctin
g B
ea
uvill
e s
urf
aces w
ith
sp
ecific
pro
pert
ies (
e.g
. re
alit
y).
� In
ve
stig
atin
g th
e m
od
uli
spa
ce
of B
ea
uvill
e s
urf
aces.