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 limits.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 group cohomologies.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. Two examples: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.
Material from the workshop
A list of participants.The workshop schedule.
A report on the workshop activities.
Papers arising from the workshop:
Cheeger constants and $L^2$-Betti numbers
by Lewis Bowen, Duke Math. J. 164 (2015), no. 3, 569-615 MR3314481
