**SUBJECT TO THE LMS AWARDING FUNDING TO THE “TRIANGLE” NEXT YEAR…**

Birkbeck, University of London, Wednesday December 13th 2017

Room B3 of 43 Gordon Square. Note that this is **not** the usual building but is nearby (if you’ve walked to Birkbeck from somewhere like Euston or King’s Cross before then you’ve practically walked past it. You’re warned that an accurate map of the building itself can be found here.) If you would like some directions then email b.fairbairn[usual funny AT symbol]bbk.ac.uk . Afterwards we will very likely go The Bree Louise followed by a curry (assuming they’re still standing – like Brexit and Winter, HS2 is coming…)

**Owing to exceptional budget constraints we politely request that any postgraduate who will be requesting that we cover their travel expenses seek-out the cheapest means of getting here possible (typically a pre-booked train is fine).**

13:30-14:30 Emilio Pierro (LSE) Big groups and their small quotients

In this talk we will discuss new techniques in determining which finite simple groups can and cannot occur as quotients of certain infinite groups. Our motivation is to prove a conjecture of Mechia-Zimmerman stating that L* _{n}*(2) is the smallest non-trivial quotient of Aut(

*F*), the automorphism groups of the free group of rank

_{n}*n*. Our techniques can also then be applied to solve the analogous conjecture of Zimmerman that S

_{2g}(2) is the smallest non-trivial quotient of MCG( Σ

_{g,b}), the mapping class group of a connected, orientable surface of genus

*g>2*with

*b*non-negative boundary components and no punctures. This is joint work with Barbara Baumeister and Dawid Kielak.

14:30-15:30 Henry Bradford (Göttingen) Short Laws for Finite Groups and Residual Finiteness Growth

A law for a group *G* is a non-trivial equation satisfied by all tuples of elements in *G*. We study the length of the shortest law holding in a finite group. We produce new short laws holding (a) in finite simple groups of Lie type and (b) simultaneously in all finite groups of small order. As an application of the latter we obtain a new lower bound on the residual finiteness growth of free groups. This talk is based on joint work with Andreas Thom.

15:30-16:00 Break

16:00-17:00 Caroline Series (Warwick) Geometry in non-discrete groups of hyperbolic isometries: Primitive stability and the Bowditch condition are equivalent

It appears that there can be geometrical conditions on a group of hyperbolic isometries which may sometimes be of interest even when the group is not discrete. We explain two different such conditions which pertain to the primitive elements in an SL(2,C) representation of the free group F_{2}. One is Minsky’s condition of primitive stability, and the other is the so-called BQ-condition introduced by Bowditch and generalised by Tan, Wong and Zhang. We prove these two conditions are equivalent, aided by an auxiliary condition which constrains the location of the axes of those primitive elements which are palindromic words.