Groups and Logic

# Tuesday, April 23, 2019 at UIC

Spring 2019 MWMT is April 23.

Speakers
: Nir Avni (Northwestern), Rizos Sklinos (Stevens), Turbo Ho (Purdue)

Schedule:
• Noon: Lunch in SEO 636.
• 1pm: Talk #1: Nir Avni
• 2:30: Talk #2: Rizos Sklinos
• 4pm: Talk #3: Turbo Ho
All talks are about an hour long including questions, in SEO 636. There will also be coffee & cookies in 636.
It is probably easiest to park in the university parking lot on Morgan St. between Roosevelt and Taylor; please bring in parking tickets (from university lots), and we can validate them. This is easiest if you bring your actual ticket with you.

The Poster

Abstracts:

Avni: Rigidity and bi-interpretability with Z for higher rank lattices.
Abstract: A lattice in a Lie group is a discrete subgroup with finite co-volume. In many contexts, there is a dichotomy between lattices in Lie groups of rank one and lattices in Lie groups of higher rank, where the two classes behave in qualitatively different ways. I will talk about this dichotomy in the context of Model Theory.

Sklinos: Fraisse constructions in the free group
Abstract: In an influential paper Fraisse obtained the ordered rationals as a limit of finite linear orders through amalgamations. Furthermore his construction implied the (ultra)-homogeneity, the countability and universality of the limit structure. Since then various adaptations of Fraisse's method had given very interesting examples in many mathematical disciplines. The random graph in graph theory and Philip Hall's universally locally finite group in group theory to name a few.
In joint work with Kharlampovich and Myasnikov we look into the possibility of applying Fraisse constructions in classes of groups that played a central role in answering Tarski's question on nonabelian free groups. In particular, we modify Fraisse's method to prove that nonabelian limit groups form a $\forall$-Fraisse class and finitely generated elementary free groups form an elementary-Fraisse class.

Ho:  Scott sentence of finitely-generated groups
Abstract: Scott showed that for every countable structure A, there is a L_{\omega_1,\omega} sentence, called the Scott sentence, whose countable models are the isomorphic copies of A. The quantifier complexity of a Scott sentence can be thought of as a measure of the complexity of the structure. Knight et al. have studied the Scott sentences of many structures. In particular, Knight and Saraph showed that a finitely-generated structure always has a \Sigma_3 Scott sentence. In this talk, we will focus on finitely-generated groups. On the one hand, most "natural" finitely-generated groups have a d-\Sigma_2 Scott sentence. On the other hand, we give a characterization of finitely-generated structures where the \Sigma_3 Scott sentence is optimal. We then give a construction of a finitely-generated group where the \Sigma_3 Scott sentence is optimal.

This is joint work with Matthew Harrison-Trainor.

Midwest Model Theory Day is supported by NSF award number 1700095.