Alexander Lubotzky, Finite Groups and Hyperbolic manifolds
Abstract: The isometry group of a closed hyperbolic n-manifold is finite. We
prove that for every n>1 and every finite group G there is an n-dimensional
closed hyperbolic manifold whose isometry group is G. This resolves a
longstanding problem whose low-dimensional case n=2 and n=3 were proved
by Greenberg ('74) and Kojima ('88) respectively. The proof is nonconstructive; it
uses a 'probabilistic method', i.e. counting results from the theory of
'subgroup growth'. The talk won't assume any prior knowledge on the
subject.
This is joint work with M. Belolipetsky.
Andrzej Zuk, Automata groups
Abstract: We will present recent developments in the theory of
automata groups.
Mark Sapir, Groups acting on tree-graded spaces
Abstract: We show that if a group acts
"nicely" on a tree-graded space, then it acts "nicely" on an R-tree.
Applications to relatively hyperbolic groups include descriptions of
relatively hyperbolic groups with infinite Out(G) and of co-Hopfian
relatively hyperbolic groups.
This is joint work with Cornelia Drutu.
Alexei Miasnikov, Asymptotic behavior of algorithmic problems in groups
Abstract: It turns out that many hard algorithmic problems are surprisingly easy on most inputs.
I will describe some recent developments in asymptotic and algorithmic group theory that lead to new efficient
algorithms. The key points of the discussion involve random van Kampen diagrams, stratification and
black holes in HNN extensions and Miller's groups.
Vladimir Shpilrain, Asymptotic density in free and free abelian groups
Abstract: Let F_k be the free group of finite rank k \ge 2 and let \alpha be the abelianization
map from F_k onto Z^k. We prove that if a
subset S of Z^k is invariant under the natural action of SL(k, Z), then the asymptotic density of S in Z^k and
the asymptotic density of its full preimage \alpha^{-1}(S) in F_k are equal. This implies, in particular, that
for every integer t = 1, the asymptotic density of the set of elements in F_k that map to t-th powers of
primitive elements in Z^k is equal to \frac{1}{t^k\zeta(k)}, where \zeta is the
Riemann zeta-function.
As an application of this result we show that the union of all proper retracts in the free group of rank two
has asymptotic density 6/\pi^2 . This contrasts with the fact that the union of all proper free factors has
asymptotic density 0.
This is joint work with I. Kapovich, I. Rivin, and P. Schupp.