Abstract
AbstractLet $$G \subsetneq \mathbb {C}$$
G
⊊
C
be a bounded, simply connected domain in $$\mathbb {C}$$
C
, and denote by $$\begin{aligned} Q(G):= \left\{ f: G \longrightarrow G \ |\ f \text { is a quasiconformal mapping of } G \text { onto } G \ \right\} \end{aligned}$$
Q
(
G
)
:
=
f
:
G
⟶
G
|
f
is a quasiconformal mapping of
G
onto
G
the quasiconformal automorphism group of G. In a canonical manner, the set Q(G) carries the structure of a (non–abelian) group with respect to composition of mappings. Moreover, we endow the set Q(G) with the topology of uniform convergence by the supremum metric $$\begin{aligned} d_{\sup }(f, g):= \sup \limits _{z \in G} \left| f(z) - g(z) \right| , f, g \in Q(G). \end{aligned}$$
d
sup
(
f
,
g
)
:
=
sup
z
∈
G
f
(
z
)
-
g
(
z
)
,
f
,
g
∈
Q
(
G
)
.
In this paper, we present results concerning topological properties of Q(G) such as completeness, separability, path–connectedness, discreteness and compactness.
Funder
Julius-Maximilians-Universität Würzburg
Publisher
Springer Science and Business Media LLC
Subject
Applied Mathematics,Geometry and Topology,Algebra and Number Theory,Analysis