Stable commutator length (scl) is a well established invariant of group elements g (write scl(g)) and has both geometric and algebraic meaning.

It is a phenomenon that many classes of non-positively curved groups have a gap in stable commutator length: For every non-trivial element g, scl(g) > C for some C>0. Such gaps may be found in hyperbolic groups, Baumslag-solitair groups, free products, Mapping class groups, etc.

However, the exact size of this gap usually unknown, which is due to a lack of a good source of “quasimorphisms”.

In this talk I will construct a new source of quasimorphisms which yield optimal gaps and show that for Right-Angled Artin Groups and their subgroups the gap of stable commutator length is exactly 1/2. I will also show this gap for certain amalgamated free products.

# Past Junior Topology and Group Theory Seminar

In a recent paper Friedl, Zentner and Livingston asked when a sum of torus knots is concordant to an alternating knot. After a brief analysis of the problem in its full generality, I will describe some effective obstructions based on Floer type theories.

Warped cones are infinite metric spaces that are associated with actions by homeomorphisms on metric spaces. In this talk I will try to explain why the coarse geometry of warped cones can be seen as an invariant of the action and what it can tell us about the acting group.

I will give a survey of known results about when two RAAGs are quasi-isometric, and will then describe a visual graph of groups decomposition of a RAAG (its JSJ tree of cylinders) that can often be used to determine whether or not two RAAGs are quasi-isometric.

If $G$ is an irreducible lattice in a semisimple Lie group, every action of $G$ on a tree has a global fixed point. I will give an elementary discussion of Y. Shalom's proof of this result, focussing on the case of $SL_2(\mathbb{R}) \times SL_2(\mathbb{R})$. Emphasis will be placed on the geometric aspects of the proof and on the importance of reduced cohomology, while other representation theoretic/functional analytic tools will be relegated to a couple of black boxes.

I will present a gentle introduction to the theory of conformal dimension, focusing on its applications to the boundaries of hyperbolic groups, and the difficulty of classifying groups whose boundaries have conformal dimension 1.