

London Mathematical Society  EPSRC Durham Symposium

Geometry and Arithmetic of Lattices

Monday 4th July  Thursday 14th July 2011


Aim and scope of the symposium
Lattices may be studied from different points of view, including
topology, geometry and number theory. In addition, research in this
area varies between the abstract and the concrete. There are many new
results from one aspect of this topic that have impact on at least
one of the others. This meeting will bring together people from a
variety of backgrounds, but who all have an interest in lattices.
By informing one another of recent progress and outlining open problems,
we hope to learn from one another to advance this area of research.
The topics covered by the conference will include:
(1) Betti numbers and covers
(2) Complex hyperbolic lattices
(3) Counting lattices
(4) Geometric properties
(5) Nonarithmeticity
(6) Reflection groups
(7) Subgroup separability of hyperbolic lattices
(8) Volumes
Here are more details of these topics.
(1) Betti numbers and covers
One of the fundamental conjectures in hyperbolic geometry states that
every hyperbolic nmanifold has a finite sheeted cover with positive
first Betti number. The 3dimensional case of this conjecture, which
is often attributed to Thurston, is widely considered as the main open
problem about the geometry and topology of closed 3dimensional manifolds,
now that Perelman has proved the Poincare and geometrisation conjectures.
There are numerous results towards different cases of the first Betti number
conjecture, which include a proof of the 3dimensional surface
subgroup conjecture for arithmetic hyperbolic 3manifolds
recently announced by Lackenby. Moreover, the question
of nonvanishing of first Betti number is very much an active question
for lattices in PU(n,1) and in some sense even more mysterious than
the real hyperbolic case. Our plan is to discuss the recent developments
with an emphasis on the higher dimensional real and complex cases.
(2) Complex hyperbolic lattices
One may construct lattices in PU(n,1), the holomorphic isometry
group of complex hyperbolic space, in several different ways. Nevertheless,
it is often hard to do so and there are relatively few explicit constructions
known. Broadly speaking, there are four major constructions: arithmetic
constructions, use of moduli of different objects, algebrogeometric
constructions based on Yau's theorem and construction of fundamental
domains. There are many connections between these approaches, and we
plan to bring together experts in different areas in order to improve
our understanding. A fake projective plane is the quotient space of
a torsionfree uniform complex hyperbolic lattice of minimal covolume.
Although the first example was given by Mumford in 1979, there is no
explicit topological description of any of these manifolds. Recently
Prasad and Yeung have classified fake projective planes. Fundamental
domains for various lattices in PU(2,1) have been constructed
recently by Deraux, Falbel, Parker and Paupert. Allcock, Carlson
and Toledo have given new examples of complex hyperbolic lattices
in high dimensions by considering the moduli space of geometric objects,
such as cubic threefolds. By bringing these researchers together we aim
to get a more unified understanding of complex hyperbolic lattices.
(3) Counting lattices
The study of distribution of lattices in Lie groups was initiated
by Lubotzky in mid 90's in connection with the congruence subgroup
problem and property $\tau$. Later on a significant progress in
this area was achieved with important contributions by Belolipetsky,
Gelander, Lubotzky, Mozes and Nikolov, to name a few. Having an
opportunity to bring together the leading experts in this rapidly
developing field, we plan to explore its possible connections with
the other topics of the symposium.
(4) Geometric properties
Many algebraic and arithmetic properties of lattices are related
with geometry of the locally symmetric spaces. This connection
provides methods and ideas for studying various geometric questions.
Using this type of approach Belolipetsky and Lubotzky showed that
every finite group can be realised as a group of isomtries of a
hyperbolic nmanifold for an arbitrary dimension $n$, which solved
an old standing problem. Another example related to this topic is
provided by the recent work of Leininger, McReynolds, Neumann, and
Reid, and Prasad and Rapinchuk on lengths of closed geodesics and
isospectral locally symmetric spaces.
(5) Nonarithmeticity
A group is called arithmetic if it is commensurable to the group
of integral points in an algebraic group defined over the rationals.
Borel and HarishChandra showed that all arithmetic groups are
lattices. Conversely, Margulis showed that all lattices in
irreducible Lie groups of rank at least two are arithmetic. Among
the rank one groups, Corlette, Gromov and Schoen showed the same
result for PSp(n,1) and F_{4(20)}. This only leaves
PO(n,1)
and PU(n,1). Nonarithmetic lattices in PO(n,1) for all values
of n were given by Gromov and PiatetskiShapiro. The first
nonarithmetic lattices in PU(2,1) were given by Mostow in 1980.
In later work with Deligne, he found more nonarithmetic lattices
in PU(2,1) and also a single example in PU(3,1). These are essentially
the only examples known. For n>3 the problem is open. Recently
Parker and Paupert have developed a new strategy that produces groups
that are candidates for being nonarithmetic lattices in PU(2,1).
Preliminary investigations indicate that this method will produce
nonarithmetic lattices that are not commensurable with any of the
previously known examples.
(6) Reflection groups
Hyperbolic reflection groups can be considered as a natural
generalisation of the classical finite and affine Weyl groups. Their
study gives rise to various interesting problems which are related to
different subjects. Our prime interest is in a recent proof of the
finiteness of the number of conjugacy classes of arithmetic maximal
hyperbolic reflection groups which was given by Agol, Belolipetsky,
Storm, and Whyte and, independently, by Nikulin following a previous
work of Maclachlan, Reid and Long. This completes a project begun
over 30 years ago by Vinberg, and opens new possibilities for classification
of the hyperbolic reflection groups. An interesting open problem is to
find analogous finiteness theorems for lattices in PU(n,1)
generated by complex reflections.
(7) Subgroup separability of hyperbolic lattices
A theme that has been very fruitful recently is the interplay between
questions about the topology of finite covers of hyperbolic manifolds
(particularly in dimension 3) and virtual injections of their fundamental
groups in right angled Artin groups or right angled Coxeter groups.
This was used by Agol, Long and Reid to prove GFERF for many
arithmetic lattices, and in extensions of their work by Bergeron, Haglund
and Wise, and by Kapovich, Potyagailo and Vinberg. The latter relates
questions of incoherence of lattices in PO(n,1) (for n>3)
and the virtual fibering question in dimension 3. In addition, work of Agol
on virtual fibering connects these questions to the topology
of 4manifolds, through work of Friedl and Vidussi on the Taubes Conjecture.
(8) Volumes
Volume is one of the principal invariants of locally symmetric spaces
and, more generally, Riemannian manifolds. In 1968 Kazhdan and Margulis
showed that any connected semisimple Lie group without compact factors
contains a lattice of minimal covolume. Since then there has been a
considerable interest in describing such extremal lattices and
computing their covolumes. This problem for the groups of isometries
of hyperbolic 3space was solved in full generality only very
recently by Gehring and Martin. More results are available if we
restrict ourselves to arithmetic lattices, for which we have volume
formulae due to Borel and Prasad. In particular, in a work of Belolipetsky
the minimal covolume arithmetic hyperbolic lattices were determined for even
dimensions, and in a recent PhD thesis of Emery this problem was solved
for the odd dimensions. This leads to a number of open questions
about geometric properties of the corresponding locally symmetric spaces
which are related to other topics in our program.

Organising Committee:
Misha Belolipetsky (Durham), Martin Bridson (Oxford), Marc Lackenby (Oxford), John Parker (Durham)

Scientific advisers:
Misha Kapovich (UC Davis), Alan Reid (UT Austin)

