### Groups and Geometry in the South East

### Property (T) and random quotients of hyperbolic groups

#### 1:30

*Calum Ashcroft (Cambridge)*

In his original manuscript on hyperbolic groups, Gromov asked whether random quotients of non-elementary hyperbolic groups have Property (T). This question was later refined by Ollivier, and then answered in the case of random quotients of free groups by Zuk (and Kotowski--Kotowski).

In this talk we answer the Gromov--Ollivier question in the affirmative. We will discuss random quotients and some of their properties, in particular with relation to Property (T).

### Connections between hyperbolic geometry and median geometry

#### 2:45

*Cornelia Drutu (Oxford)*

In this talk I shall explain how groups endowed with various forms of hyperbolic geometry, from lattices in rank one simple groups to acylindrically hyperbolic groups, present various degrees of compatibility with the median geometry. This is joint work with Indira Chatterji, and with John Mackay.

### TEA

#### 3:45

### Division, group rings, and negative curvature

#### 4:00

*Grigori Avramidi (Bonn)*

In 1997 Delzant observed that fundamental groups of hyperbolic manifolds with large injectivity radius have nicely behaved group rings. In particular, these rings have no zero divisors and only the trivial units. In this talk I will explain how to extend this observation to show such rings have a division algorithm (generalizing the division algorithm for group rings of free groups discovered by Cohn) and that these group rings have``freedom theorems’’ showing that all of their ideals that are generated by few elements are free, where the specific value of `few’ depends on the injectivity radius of the manifold (which can be viewed as generalizations from subgroups to ideals of some freedom theorems of Delzant and Gromov). This has geometric consequences to the homotopy classification of 2-complexes with surface fundamental groups and to complexity of cell structures on hyperbolic manifolds.