Title: Subgroups of almost finitely presented groups
Abstract: Almost finitely presented groups, or groups of type FP_2, lie somewhere between finitely generated groups and finitely presented groups. Until recently they were thought to be a lot closer to finitely presented groups. I will describe a recent result in which they behave more like arbitrary finitely generated groups.
Title: Deligne-Mostow Lattices and cone metrics on the sphere
Abstract: Finding lattices in PU(n,1) has been one of the major challenges of the last decades. One way of constructing lattices is to give a fundamental domain for its action on the complex hyperbolic space.One approach, successful for some lattices, consists of seeing the complex hyperbolic space as the configuration space of cone metrics on the sphere and of studying the action of some maps exchanging the cone points with same cone angle. In this talk we will see how this construction can be used to build fundamental polyhedra for Deligne-Mostow lattices with 2- and 3-fold symmetry.
Title: The Heisenberg group has rational growth in all generating sets
Abstract: Given a group G and a finite generating set \calG the (spherical) growth function f_\calG(x)=a_0+a_1x+a_2x^2+… is the series whose coefficients an count the number of group elements at distance n from the identity in the Cayley graph Γ_\calG(G). For hyperbolic groups and virtually abelian groups, this is always the series of a rational function regardless of generating set. Many other groups are known to have rational growth in particular generating sets. In joint work with Moon Duchin, we show that the Heisenberg group also has rational growth in all generating sets. The first ingredient in this result is to compare the group metric, which we can see as a metric on the integer Heisenberg group with a metric on the real Heisenberg group. This latter is induced by a norm in the plane which is in turn induced by a projection of the generating set. The second ingredient is a wondrous theorem of Max Bensen regarding summing the values of polynomical over sets of lattice points in families of polytopes. We are able to bring these two ingredients together by showing that every group element has a geodesic whose projection into the plane fellow-travels a well-behaved set of polygonal paths.
Title: Relative automorphisms of right-angled Artin groups
Abstract: We look at the group of outer automorphisms of a right-angled Artin group preserving a set of special subgroups (i.e. subgroups coming from subgraphs of the defining graph). I will give some examples to show how these groups arise naturally in the study of the whole automorphism group, and sketch how such groups are finitely generated, and up to finite index, have finite classifying spaces. This is joint work with Matthew B Day.