**Speaker:** Colva Roney-Dougal (St Andrews)

**Title:** *Random Presentations with all Relators of Length k*

**Abstract:**

Gromov defined a density model of random groups as follows. Fix a density d between 0 and 1, integers n and k both at least 2, and consider the presentation with n generators, and \((2n-1)^{kd}\) random relators, each freely cyclically reduced and of length k. Gromov showed that for any fixed n, as k tends to infinity, then asymptotically almost surely the group presented is cyclic of order dividing two when d is greater than 1/2, and is infinite and hyperbolic when d is less than 1/2. I will survey some related results, and present some recent joint work with Calum Ashcroft, where we instead fix k, and allow n to tend to infinity.