# Symmetry in Newcastle

## Monday, 10th May 2021

V205, Mathematics Building

# Profesor Titular Yago Antolin

(ICMAT, Universidad Complutense de Madrid)

# Geometry and Complexity of positive cones in groups

A positive cone on a group $G$ is a subsemigroup $P$, such that $G$ is the disjoint union of $P$, $P^{-1}$ and the trivial element. Positive cones codify naturally $G$-left-invariant total orders on $G$. When $G$ is a finitely generated group, we will discuss whether or not a positive cone can be described by a regular language over the generators and how the ambient geometry of $G$ influences the geometry of a positive cone. This will be based on joint works with Juan Alonso, Joaquin Brum, Cristobal Rivas and Hang Lu Su.

# Dr Robert Kropholler

(Mathematisches Institut, WWU Münster)

# Groups of type $FP_2$ over fields but not over the integers

Being of type $FP_2$ is an algebraic shadow of being finitely presented. A long standing question was whether these two classes are equivalent. This was shown to be false in the work of Bestvina and Brady. More recently, there are many new examples of groups of type $FP_2$ coming with various interesting properties. I will begin with an introduction to the finiteness property $FP_2$. I will end by giving a construction to find groups that are of type $FP_2(F)$ for all fields $F$ but not $FP_2(Z)$.