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 π -->
1
(
X
,
t
)
⋂ ⋂ -->
n
∈ ∈ -->
N
K
e
r
ϝ 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 π -->
1
(
X
,
t
)
{\displaystyle \pi _{1}(X,t)}
is the fundamental group of X and
ϝ ϝ -->
:
π 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 ϝ -->
n
:
π 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