Crazy Circles !
This figure
is a planar circle diagram for the icosahedral graph D
shown below. The graph D has a vertex for each circle in the diagram and an edge
for each pair of non-intersecting circles. This diagram can be used to embed a
hyperbolic 3-manifold group as a subgroup of a surface mapping class group and,
using the fact that the diagram is planar, as a subgroup of a braid
group. The idea is sketched below, and is explained in detail in my recent
preprint with Bert Wiest –
The
icosahedral graph D
; and a right-angled hyperbolic dodecahedron
Let W
denote the group generated by the reflections in the faces of a regular
right-angled dodecahedron in hyperbolic 3-space. This is a discrete subgroup of
Isom(H3) – the isometry group of hyperbolic 3-space. The group W is also
the right-angled Coxeter group W(D)
associated to the icosaedral graph D (namely, it is generated by involutions, one for
each vertex of D,
and defined by the relations that any two involutions s, t commute (st=ts) if
the corresponding vertices of D
span an edge). One can find finite index subgroups of W (for example, the
commutator subgroup) which also sit naturally as subgroups of the right-angled
Artin group (or graph group) G(D)
associated to the graph D. (This latter group is generated by a number of
elements corresponding to the vertices of
D and is defined by commuting relations as for the
Coxeter group. However, we do not
suppose that the generators are involutions and, unlike the Coxeter group, the Artin
group has no elements of finite order). These torsion free finite index
subgroups of W are fundamental groups of hyperbolic 3-manifolds which can be
built by gluing together a finite number of copies of the hyperbolic
dodecahedron along their faces.
Finally we
can use the circle diagram to embed the group G(D),
and hence each of these 3-manifold groups into the group of diffeomorphisms of
the plane (and subsequently into a braid group) by representing each generator
as a diffeomorphism which fixes everything outside a very small neighbourhood
of the corresponding circle in the diagram, and is sufficiently complicated
inside the neighbourhood of the circle.