I will give a brief introduction to laminar groups which are groups of orientation-preserving homeomorphisms of the circle admitting invariant laminations. The term was coined by Calegari and the study of laminar groups was motivated by work of Thurston and Calegari-Dunfield. We present old and new results on laminar groups which tell us when a given laminar group is either fuchsian or Kleinian.

For an infinite, residually finite group, it is interesting to ask what properties of the group are captured by its finite quotients. We will discuss how to use ideas of Bridson-McReynolds-Reid-Spitler to show, for example, that the fundamental group of zero surgery on the knot 6_2 is completely determined (among all residually finite groups) by the collection of its finite quotients.

We use topological methods to solve special cases of a fundamental problem in group theory, the Kervaire conjecture, which has connection to various problems in topology. The conjecture asserts that, for any nontrivial group G and any element w in the free product G*Z, the quotient (G*Z)/<> is still nontrivial. We interpret this as a problem of estimating the minimal complexity (in terms of Euler characteristic) of surface maps to certain spaces. This gives a conceptually simple proof of Klyachko's theorem that confirms the Kervaire conjecture for any G torsion-free. We also obtain new results concerning injectivity of the map G->(G*Z)/<> when w is a proper power.

In this talk I will give some purely topological construction for hyperbolic 3-manifolds with infinitely generated fundamental group, this will let us constructs many infinite-type hyperbolic 3-manifolds. Then, I will say something about the set of hyperbolic structures that they can admit.

In this talk, we show that most compactly supported big mapping class groups cannot be realized as a subgroup of the homeomorphism group. Time permitting, we will also prove the non-realizability of the mapping class group of the plane minus a Cantor set or the sphere minus a Cantor set. This is (ongoing) joint work with Lei Chen.

We will discuss about smooth actions on manifolds by lattices in $\textrm{SL}(n,\mathbb{R})$ with $n\ge 3$. The Zimmer program aims to “classify” such actions motivated by superrigidity results by Margulis and Zimmer. We expect to understand the manifold or the lattice from the action unless the action is nontrivial. When the dimension is at most n-1, recent works by Brwon-Fisher-Hurtado and Brown-Rodriguez Hertz-Wang gave a complete answer. In this case, we only see zero entropy actions; isometric or projective actions. In this talk, we give partial answers for actions on the “n-dimensional” manifold with “positive entropy” by lattices in $\textrm{SL}(n,\mathbb{R})$, $n\ge 3$. Part of the talk is ongoing work with Aaron Brown.

Feighn and Handel constructed CTs, a powerful tool for studying outer automorphisms of free groups, in 2011. Recently, I extended their work to the case of outer automorphisms of free products of groups. In this talk we will aim to understand the information carried by a CT and highlight some of the similarities and differences between the free group case and the general free product case.

Gromov defined a notion of boundary at infinity for \delta-hyperbolic scapes which has turned out to be an essential tool in the study of the large scale geometry of these groups. We attempt to find the natural analogue of the Gromov boundary for a larger class of metric spaces. We start in the setting of metric spaces that resemble the geometry of relatively hyperbolic groups. We define a notion of a boundary that is compact, metrizable and invariant under quasi-isometries, furthermore, it contains the sub-linearly Morse boundary as a dense topologic subspace. When the space is in fact a relatively hyperbolic group, this boundary coincides with the Bowditch boundary. This is a joint work (in progress) with Yulan Qing and Giulio Tiozzo.

We describe when a big mapping class group has a conjugacy class which is dense, somewhere dense or meager. Our techniques are based on the work of Truss, Kechris and Rosendal.

Relatively hyperbolic groups generalize geometrically finite Kleinian groups acting on real hyperbolic space $H^3$. The boundaries of relatively hyperbolic groups generalize the limit sets of geometrically finite Kleinian groups. Since the boundary of $H^3$ is $S^2$, the limit set of every Kleinian group is planar. Can every relatively hyperbolic group with planar boundary be realized as a Kleinian group? The answer is no, and we will give illustrative examples to show the many ways this can fail. However, we will show that in the absence of cut points in the boundary, such groups have the property that all their peripheral subgroups are surface groups. Under additional hypotheses, we outline a proof that relatively hyperbolic groups with planar boundary can be realized as Kleinian groups. This is joint work with Chris Hruska.