L2 invariants and their relatives for finitely generated groups
September 12 to September 16, 2011
American Institute of Mathematics,
Palo Alto, California
and Dimitri Shlyakhtenko
This workshop, sponsored by
AIM and the
will be devoted to the study of the
asymptotic behavior of some natural invariants of finitely generated
groups. Topics addressed at the meeting include L2-Betti numbers, rank
gradient, free entropy dimension, measurable actions of groups and cost,
graph limits and graphings, groups acting on rooted trees, mod p homology
and cohomology and probabilistic aspects of infinite groups. Since the
topics span several large areas of mathematics, the idea is to invite
representatives of these areas who are interested in neighbor areas and are
willing to assimilate each others point of view.
The main directions for the proposed workshop are the following.
A list of invariants. There is an array of invariants of
great interest in various fields that study groups: the free entropy
dimension (von Neumann algebras), the first L2 Betti number (topology
and L2-cohomology), the cost (ergodic theory, percolation and graph
limits), the rank gradient and the mod p homology growth (asymptotic group
theory) and the growth of the Heegaard genus along a covering tower of a 3-manifold (3-manifold theory). It is known that -- whenever these
invariants make sense -- they can be arranged in a list on inequalities and
for a fairly large class of groups, including amenable groups and groups
close to free groups, equality is known all the way. There is no
counterexample known yet!
Approximating by finite structures A finitely generated
group is sofic, if its Cayley graph is a limit of finite graphs in the local
convergence topology. All amenable groups are sofic and until now, there is
no group known that is not. The Connes embedding problem asks the same
question, just instead of finite permutations, we want to approximate the
group by finite dimensional matrices, using the trace norm. From yet another
direction, there is an emerging analytic theory of finite graph sequences of
bounded degree that is connected to these topics. For instance, the most
natural way to express Luck approximation is using the language of graph
Free probability theory. Questions around the value of
free entropy dimension are closely related to Connes embedding problem;
indeed, free entropy dimension involves a qualitative study of the
"left number" of approximations possible for a given group. All
the existing lower estimates can be obtained in a probabilistic way using
the theory of free stochastic differential equations whose coefficients are
related to objects from L2 cohomology. Analytic questions about
stationarity of solutions end up being probabilistic questions about certain
processes on the discrete group, which may be interesting and could possibly
be resolved using geometric ideas.
von Neumann algebras. This subject has seen a number of
amazing recent advances, thanks to the work of Popa, Ozawa and others. One
of the chief questions is the extent to which the von Neumann algebra of a
group depends on the underlying group. "Experimentally", the answer seems to
be rather related to the question of when two groups are measure-equivalent,
i.e., when can they induce the same orbit equivalence relation. Thus there
is a strong connection between general von Neumann algebra theory and the
theory of orbit equivalence relations. Group-geometric ideas have been
crucial in many works in von Neumann algebra theory. For example, there
appears to be a connection between Popa's notion of deformation of a von
Neumann algebra (that plays a key role in his work) and various kinds of
Chains of subgroups and actions on rooted trees. A
finitely generated, residually finite group acts by automorphisms on coset
trees (these ate locally finite rooted trees). These actions extend to a
measure preserving action on the boundary of the tree. This allows one to
play the following three against one another: the dynamics of this boundary
action, asymptotic properties of the chain that gives the coset tree and
properties of the infinite group.
1. Under mild conditions, the spectral measure of the Markov operator on
finite quotients converges to the spectral measure of the Markov operator.
In some cases, this allows one to compute the spectral measure.
2. The rank gradient of the chain can be expressed from the cost of the
boundary action. It is not known whether the rank gradient depends on the
chain; both possible answers would solve a distinguished problem, one in 3-manifold theory and the other in topological dynamics.
Spanning forests, cost and percolation. Random spanning
forests of lattices are widely investigated in probability theory, because
of their connection to random walks and percolation. Maybe the most
transparent way to introduce the first L2 Betti number of a group is to
use the expected degree of a free spanning forest on its Cayley graph.
Recently, minimal spanning forests were used to partially solve the Diximier
problem that asks whether non-amenable groups are necessarily
non-unitarizable. On the other hand, the best known general estimates
between the critical values of edge percolation of a Cayley graph derive
from the cost of i.i.d.
The workshop will differ from typical conferences in some regards.
Participants will be invited to suggest open problems and questions
before the workshop begins, and these will be posted on the workshop
website. These include specific problems on which there is hope of
making some progress during the workshop,
as well as more ambitious problems which may influence
the future activity of the field.
Lectures at the
workshop will be focused on familiarizing the participants with the
background material leading up to specific problems, and the
schedule will include discussion and parallel working sessions.
The deadline to apply for support to participate in this
workshop has passed.
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