Uri Andrews (Wisconsin)

Computability of the Model Theory of Groups

Model theory gives a characterization of modular strongly minimal expansions of groups as quasi-vector spaces over a division ring. By carefully considering this analysis in a computable way, we derive results about computable presentations of these structures and the complexity of their theories, answering questions in computable model theory. This more careful analysis requires a reworking of some tools from both model theory and basic linear algebra. (Joint work with Alice Medvedev)

Laura Ciobanu (Fribourg)

Three decisions problems in free groups and their asymptotic behavior

In a free group $F$ of finite rank one can formulate the automorphism, endomorphism and monomorphism problems as follows.

Is there an algorithm that determines whether there exists a homomorphism φ : F → F such that for any given u and v in F one has φ(u)=v?

If φ is required to be an automorphism, this is called the automorphism problem, and the answer is affirmative by Whitehead's algorithm. The endomorphism problem simply requires φ to be an endomorphism, and is equivalent to the satisfiability of homogeneous equations in free groups, which is settled by Makanin's algorithm. We will show that this problem is also solvable in the case when φ is required to be a monomorphism. This is joint work with A. Ould Houcine.

We will then discuss the asymptotic behaviour of these questions, that is, how likely it is for a word to be a

φ-image of another, for some homomorphism φ. In particular, we will show that the satisfiability of homogeneous equations in surface groups is an intermediate property. This is joint work with Y. Antolin and N. Viles.

Isaac Goldbring (UCLA)

Ends of groups from a nonstandard perspective.

An important geometric invariant of a finitely generated group is its space of ends. The space of ends of an arbitrary topological space may be intuitively described as the set of "path components at infinity."  For proper geodesic spaces, I show how to use the language of nonstandard analysis to make the aforementioned heuristic precise. When this nonstandard characterization is applied to the case of a Cayley graph of a finitely generated group, it becomes easier to perform calculations and prove theorems, as will be illustrated through a few examples. I will end the talk with some ideas for future applications.

Masato Mimura (Tokyo)

Property (TT)/T and homomorphism superrigidity into mapping class groups

We see the following: every homomorphism from finite index subgroups of a universal lattices to mapping class groups of orientable surfaces (possibly with punctures), or to outer automorphism groups of finitely generated nonabelian free groups must have FINITE image, where the universal lattice denotes the special linear group Γ=SLm(Z[x1,...,xk]) with m at least 3 and k finite. Also we show the same results for symplectic universal lattices Sp2m(Z[x1,...,xk]) with m at least 2.  Furthermore, a certain measure equivalence analogue of them is also treated. These results can be regarded as a non-arithmetization of superrigidity of Farb--Kaimanovich--Masur and Bridson--Wade, at least for any lattices  (including COCOMPACT ones) in algebraic groups of the form of SLn, n at least 3; of Sp2n, n at least 2; or their product groups. To show the statements above, we introduce a notion of  “property (TT)/T", which is a strengthening of property (T) of Kazhdan and is a weakening of property (TT) of Monod.

Christian Rosendal (UIC)

Complete metric groups acting on trees

We consider actions of completely metrisable groups on simplicial trees in the context of

the Bass–Serre theory. Our main result characterises continuity of the amplitude function corresponding

to a given action. Under fairly mild conditions on a completely metrisable group G , namely, that the

set of elements generating a non-discrete or finite subgroup is somewhere dense, we show that in any

decomposition as a free product with amalgamation, G = A *C B , the amalgamated groups A, B and C

are open in G .

Thomas Scanlon (Berkeley)

(Non-)bi-interpretability with arithmetic

No infinite finitely generated ring is categorical in any cardinal, but relative to this class every such ring is categorical. The key observation required to prove this result is that every (infinite) finitely generated integral domain is bi-interpretable with arithmetic and every finitely generated (infinite) ring is very nearly so.  In this talk, I will focus on some paradigmatic examples, specifically, Z x Z and  Z[ε]/(ε2).  The question of bi-interpretability with arithmetic in the case of the dual numbers comes down to the questions of whether there exist non-trivial derivations on nonstandard models of arithmetic.  (joint with M. Aschenbrenner, A. Khélif and E. Naziazeno)

Henry Wilton (UCL)

One-ended subgroups of limit groups

A longstanding question in geometric group theory asks whether every hyperbolic group with the property that every subgroup of infinite index is free is the fundamental group of a surface. This question is still open for some otherwise well-understood classes of groups. In this talk, I will explain why the answer is affirmative for limit groups.  The hardest case is that of graphs of free groups with cyclic edge groups. I will also discuss the extent to which these techniques help with the harder problem of finding surface subgroups.