Introduction to Teichmüller theory, old and new, II

44 Athanase Papadopoulos
that there is a natural correspondence between homotopy classes of homeomorphisms
between finite covers of a surface and elements of the virtual automorphism group
of the fundamental group of that surface. A related natural object of study is the
baseleaf preserving mapping class group of the compact solenoid S, defined (modulo
some technicalities) after the choice of a baseleaf, as the group of isotopy classes of
baseleaf preserving self-homeomorphisms of this space S. C. Odden proved in 2004
that the baseleaf preserving mapping class group of S is naturally isomorphic to the
virtual automorphism group of the fundamental group of the base surface. This result
is considered as an analogue of the Dehn–Nielsen–Baer Theorem that describes the
mapping class group of a closed surface of genus ≥ 1 as the outer automorphism
group of its fundamental group. Markovic & Šarić proved that the baseleaf preserving
mapping class group of the solenoid does not act discretely on T (S), a result which
should be compared to the fact that in general, the mapping class group of surfaces of
infinite type does not act discretely on the corresponding Teichmüller space.
The non-compact solenoid, also called the punctured solenoid, and denoted by Snc,
is defined in analogy with the compact solenoid, as the inverse limit of the system of all
pointed finite sheeted coverings of a base surface S0 of negative Euler characteristic,
except that here, S0 is a punctured surface. A study of the noncompact solenoid was
done by Penner & Šarić, who equipped that space with the various kinds of structures
that exist on the compact solenoid, namely, complex structures, quasiconformal maps
between them, a Teichmüller space, and a mapping class group which is isomorphic to
a subgroup of the commensurator group of the base surface preserving the peripheral
structure (in analogy with the case of the mapping class group of a punctured surface).
Chapter 19 of this Handbook, written by Dragomir Šarić, contains a review of the
theory of the compact solenoid and of recent work on the noncompact solenoid Snc
by Penner & Šarić, as well as work by Bonnot, Penner and Šarić on a cellular action
of the mapping class group of Snc. In analogy with the corresponding situation for
punctured surfaces, there is a decorated Teichmüller space of the noncompact solenoid,
with associated λ-length coordinates, and a convex hull construction of fundamental
domains which gives an interesting combinatorial structure for this Teichmüller space,
generalizing an analogous structure that was developed by Penner for the Teichmüller
space of a punctured surface. An explicit set of generators for the mapping class
group of the noncompact solenoid is also discussed. Note that no such explicit set of
generators for the compact solenoid is known. It is conjectured that the mapping class
groups of the compact and of the noncompact solenoids are not finitely generated.
Chapter 19 ends with a discussion of open problems on the Teichmüller space and
on the mapping class group of the compact and the noncompact solenoids.