A surface of genus 2
This figure
describes a closed orientable surface of genus 2 –
the surface is built by gluing together 12 squares as indicated (the edges
around the boundary of the figure are glued together in pairs as indicated by
the labels). The arrows and labels on the edges in the figure describe a homomorphism
from the fundamental group G of the genus 2 surface to the right-angled
Artin group given by the presentation
< a, b, c, d, e, f | ab=ba, bc=cb,
ac=ca, ad=da, be=eb, cf=fc, de=ed, ef=fe, df=fd
>
which
is associated to the triangular prism graph. It can be shown that this
homomorphism is injective (in fact a quasi-isometric embedding of groups). It
follows that every orientable hyperbolic surface
group may be embedded as a subgroup of the triangular prism right-angled Artin group, answering (in the negative) a question
appearing in the paper
C. McA. Gordon, D. Long, A. Reid, Surface
subgroups of Coxeter and Artin
groups, (Question 3.3).
This shows
that the question of exactly which right-angled Artin
groups admit closed hyperbolic surface subgroups is rather more delicate than
at first imagined. This and other examples of its kind will be described in a
joint paper with Michah Sageev
and Mark Sapir (currently in preparation but hopefully
to appear soon).