Alladi Ramakrishnan Hall

Elementary equivalence in some classes of geometric groups: Artin groups of finite type and mapping class groups of closed surfaces

T V H Prathamesh

ISI Chnennai

Elementary theory of a group G refers to all first-order sentenences which hold true in the group G. It can be understood as a
generalization of properties of groups, which are definable in terms of polynomials. We say that two groups are elementarily equivalent if they have the same elementary theory.

Irreducible Artin groups of finite type can be parametrized via their associated Coxeter diagrams into six sporadic examples and four
infinite families, each of which is further parametrized by the natural numbers. Within each of these four infinite families, we
investigate the relationship between elementary equivalence and isomorphism. For three out of the four families, we show that two
groups in the same family are equivalent if and only if they are isomorphic; a positive, but weaker, result is also attained for the
fourth family. In particular, we show that two braid groups are elementarily equivalent if and only if they are isomorphic.

As a consequence of our work, we prove that there are infinitely many elementary equivalence classes of irreducible Artin groups of
finite type. We also show that mapping class groups of closed surfaces - a geometric analogue of braid groups - are elementarily equivalent if and only if they are isomorphic.

No prerequisites in logic or artin groups will be assumed.