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In
geometric group theory and
dynamical systems the iterated monodromy group of a
covering map is a
group describing the
monodromy action of the
fundamental group on all
iterations of the covering. A single covering map between spaces is therefore used to create a tower of coverings, by placing the covering over itself repeatedly. In terms of the
Galois theory of covering spaces , this construction on spaces is expected to correspond to a construction on groups. The iterated monodromy group provides this construction, and it is applied to encode the combinatorics and
symbolic dynamics of the covering, and provide examples of
self-similar groups .
Definition
The iterated monodromy group of f is the following
quotient group :
I
M
G
f
:=
π
1
(
X
,
t
)
⋂
n
∈
N
K
e
r
ϝ
n
{\displaystyle \mathrm {IMG} f:={\frac {\pi _{1}(X,t)}{\bigcap _{n\in \mathbb {N} }\mathrm {Ker} \,\digamma ^{n}}}}
where :
f
:
X
1
→
X
{\displaystyle f:X_{1}\rightarrow X}
is a
covering of a
path-connected and
locally path-connected topological space X by its subset
X
1
{\displaystyle X_{1}}
,
π
1
(
X
,
t
)
{\displaystyle \pi _{1}(X,t)}
is the
fundamental group of X and
ϝ
:
π
1
(
X
,
t
)
→
S
y
m
f
−
1
(
t
)
{\displaystyle \digamma :\pi _{1}(X,t)\rightarrow \mathrm {Sym} \,f^{-1}(t)}
is the
monodromy action for f .
ϝ
n
:
π
1
(
X
,
t
)
→
S
y
m
f
−
n
(
t
)
{\displaystyle \digamma ^{n}:\pi _{1}(X,t)\rightarrow \mathrm {Sym} \,f^{-n}(t)}
is the monodromy action of the
n
t
h
{\displaystyle n^{\mathrm {th} }}
iteration of f ,
∀
n
∈
N
0
{\displaystyle \forall n\in \mathbb {N} _{0}}
.
Action
The iterated monodromy group acts by
automorphism on the
rooted tree of preimages
T
f
:=
⨆
n
≥
0
f
−
n
(
t
)
,
{\displaystyle T_{f}:=\bigsqcup _{n\geq 0}f^{-n}(t),}
where a vertex
z
∈
f
−
n
(
t
)
{\displaystyle z\in f^{-n}(t)}
is connected by an edge with
f
(
z
)
∈
f
−
(
n
−
1
)
(
t
)
{\displaystyle f(z)\in f^{-(n-1)}(t)}
.
Examples
Iterated monodromy groups of rational functions
Let :
If
P
f
{\displaystyle P_{f}}
is finite (or has a finite set of
accumulation points ), then the iterated monodromy group of f is the iterated monodromy group of the covering
f
:
C
^
∖
f
−
1
(
P
f
)
→
C
^
∖
P
f
{\displaystyle f:{\hat {C}}\setminus f^{-1}(P_{f})\rightarrow {\hat {C}}\setminus P_{f}}
, where
C
^
{\displaystyle {\hat {C}}}
is the
Riemann sphere .
Iterated monodromy groups of rational functions usually have exotic properties from the point of view of classical group theory. Most of them are infinitely presented, many have
intermediate growth .
IMG of polynomials
The
Basilica group is the iterated monodromy group of the polynomial
z
2
−
1
{\displaystyle z^{2}-1}
See also
References
Volodymyr Nekrashevych,
Self-Similar Groups , Mathematical Surveys and Monographs Vol. 117, Amer. Math. Soc., Providence, RI, 2005;
ISBN
0-412-34550-1 .
Kevin M. Pilgrim, Combinations of Complex Dynamical Systems , Springer-Verlag, Berlin, 2003;
ISBN
3-540-20173-4 .
External links