# Interpretations, back-and-forth games, and group von Neumann algebras

## Matthew Harrison-Trainor

When defining the Ehrenfreucht-Fraisse back-and-forth games, it is common for model theorists to say that each player plays a single element at a time, while many computability theorists will often say that each player can play a tuple of arbitrary length. Both versions of these games appeared in Ehrenfreuchtâ€™s first treatment of back-and-forth games. However the two versions of the games can behave very differently, in particular by how they transfer through constructions like the construction from a ring \(R\) of the polynomial ring \(R[x]\). I will talk about some interesting aspects about the differences between the two versions of the back-and-forth games, and when one might use each one. One interesting example, from joint work with Isaac Goldbring, is the construction from a group of its group von Neumann algebra.