Fall 2009 Topics Course -- Group actions on cube complexes
Geometric group theory seeks to understand properties of groups using their
actions on spaces with nice geometric properties. One such property is that
of negative, or, more generally, nonpositive curvature, which has proven
useful in understanding the structure of discrete subgroups of semisimple Lie
groups and 3-manifolds, to name a few of many examples. The local property of
nonpositive curvature imposes strong restrictions on the global topology of a
Riemannian manifold; for example, the Cartan-Hadamard theorem asserts that its
universal cover is homeomorphic to R^n.
In this course we will consider cube complexes -- cell complexes with cells
of the form I^n and attaching maps that restrict to isometries of faces. The
Euclidean metric on I^n determines a metric on a cube complex for which many
geometric features, notably a coarse version of nonpositive curvature known as
the CAT(0) condition, can be readily discerned from its combinatorial
structure. Many interesting groups act on CAT(0) cube complexes;
for instance,
my own motivation for an interest in cube complexes is the fact that
the fundamental groups of many finite-volume hyperbolic
3-manifolds act faithfully, properly, and cocompactly on CAT(0) square
complexes. Exactly which 3-manifold groups have similar properties is a
subject of current research interest. Other
examples which we will study are free groups and right-angled Artin groups.
This course should be of interest to any student in
low-dimensional topology or geometric group theory. Prerequisites are
an understanding of abstract algebra and algebraic topology at the first-year
graduate level. A rough course outline is as follows.
1. Basic definitions and geometric properties; for example, completeness,
geodesics, nonpositive curvature (the CAT(0) condition) and flats.
Structure theorems (eg, Cartan-Hadamard) and consequences for groups.
2. Fundamental examples of group actions on cube complexes: free groups
(a tree is a 1-dimensional cube complex), free abelian groups, and
right-angled Artin groups (RAAGs).
3. Actions from splittings: The well known ``Fundamental theorem of
Bass-Serre theory'' asserts that a group acts nicely on a tree if and only
if it has a ''splitting'' which decomposes it into subgroups. We will
describe this theory and an extension due to Sageev, which associates an
action on a CAT(0) cube complex to an ''immersed splitting''.
4. Quasiconvex subgroups. A subset S of a metric space X is quasiconvex if
every geodesic in X between points of S stays close to S (it is convex if
such geodesics are actually contained in S, but quasiconvexity is a more
useful notion in our setting). A subgroup of a group acting on a cube
complex is quasiconvex if the orbit of some point is. We will describe
examples and nonexamples of quasiconvex subgroups, prove the following
theorem: quasiconvex subgroups of free groups and RAAGs are "virtual
retracts'', and describe why it is useful.
5. Discrete Morse Theory. If time permits, we will introduce a powerful tool
used by Bestvina-Brady (see below), among others, to describe unexpected
properties of some groups acting on cube complexes.
There will not be a textbook, although some basic material will likely be
lifted from Bridson-Haefliger's ``Metric spaces of non-positive curvature".
Other sources may include the following papers.
M. Bestvina; N. Brady, ``Morse theory and finiteness properties of
groups", Invent. Math. 129 (1997), no. 3, 445-470.
M. Gromov, ``Hyperbolic groups", Essays in group theory (S.M. Gersten,
ed), Springer Verlag, MSRI Pub. 8 (1987), 75-263.
F. Haglund; D. Wise, ``Special cube complexes", Geom. Funct. Anal. 17
(2008), no. 5, 1551-1620.
M. Sageev, ``Ends of group pairs and non-positively curved cube
complexes", Proc. London Math. Soc. (3) 71 (1995), no. 3, 585-617.