An important class of Golod-Shafarevich groups consists of the fundamental groups of compact hyperbolic 3-manifolds or, equivalently, torsion-free lattices in SO(3,1). In 1983, Lubotzky used this fact to prove that arithmetic lattices in SO(3,1) do not have the congruence subgroup property. More recently, Lubotzky and Zelmanov proposed a group-theoretic approach (based on Golod-Shafarevich techniques) to an even more ambitious problem, Thurston's virtual positive Betti number conjecture. This approach led to the following question: is it true that Golod-Shafarevich groups never have property (\tau)? I will show that the answer to the above question is negative in general and describe examples of Golod-Shafarevich groups with property (\tau) (in fact, property (T)) which are given by lattices in certain topological Kac-Moody groups over finite fields.
Let L be the direct sum of the complex Leech lattice and a hyperbolic cell.
We will describe 26 complex reflections of order 3 generating the automorphism group of L that form the same Coxeter diagram D under braiding and commuting relations. The automorphism group acts on the Complex hyperbolic space CH^{13}. A conjecture due to Daniel Allcock says that the bimonster should be a quotient of the orbifold fundamental group of (CH^{13} \ {Mirrors of Reflection})/Aut(L).
We'll see that our example has surprising analogies with the theory of Weyl group that make our proofs work. D acts as the Coxeter-Dynkin diagram for the reflection group of L. There is a parallel story for the quaternionic Leech lattice where the surprises repeat.
Motivated by Sageev's correspondence between G-cubings and codimension-1 subgroups of G, we show how an invariant system H of convex halfspaces in a CAT(0) G-space X allows for a better understanding of the topology of Bd(X) - the visual boundary of X.
To every cubing C we associate a combinatorial boundary RC. Given G,X and H as above we form the (dual) Sageev cubing C(H) and use its boundary RC(H) to construct a stratification of Bd(X). Properties of C(H) -- e.g., its properness, co-compactness -- disclose important information about G -- e.g., biautomaticity, by Niblo-Reeves. Exploiting an analogy with Coxeter groups and their parallel walls properties defined by Davis-Shapiro we address the properness and co-compactness issues using the stratification of Bd(X).
We also discuss what kind of topological information is encoded in our stratification of Bd(X), and formulate some questions. Time permitting, we will discuss plausible applications to JSJ splittings of CAT(0) groups.
We will give a new approach, and a generalization, of classical theorems of Ferrand, Obata and Schoen, showing that very weak conditions on the dynamics of the automorphism group of such structures, implies very strong rigidity results on the geometry.