Related papers: Orderings of mapping class groups after Thurston

There is a family of hyperbolic groups known as hyperbolic hydra which contain heavily distorted free subgroups. We prove the existence of Cannon--Thurston maps (that is, maps of the boundaries induced by subgroup inclusion) for these free…

In this note we show that many subgroups of mapping class groups of infinite-type surfaces without boundary have trivial centers, including all normal subgroups. Using similar techniques, we show that every nontrivial normal subgroup of a…

We prove a result that relates the number of homomorphisms from the fundamental group of a compact nonorientable surface to a finite group $G$, where conjugacy classes of the boundary components of the surface must map to prescribed…

This note studies the Burnside problem for homeomorphism groups of compact connected manifolds. For surfaces, we prove that the identity component of the homeomorphism group is torsion-free precisely when the surface is not the sphere,…

We develop the theory of Thurston maps that are defined everywhere on the topological sphere $S^2$ with a possible exception of a single essential singularity. We establish an analog of the celebrated W. Thurston's characterization theorem…

These revised lecture notes are an expository account of part of the proof of Thurston's Ending Lamination Conjecture for Kleinian surface groups, which states that such groups are uniquely determined by invariants that describe the…

We study groups of C^1 orientation-preserving homeomorphisms of the plane, and pursue analogies between such groups and circularly-orderable groups. We show that every such group with a bounded orbit is circularly-orderable, and show that…

We give a classification and complete algebraic description of groups allowing only finitely many (left multiplication invariant) circular orders. In particular, they are all solvable groups with a specific semi-direct product…

We give a finite presentation of the mapping class group of an oriented (possibly bounded) surface of genus greater or equal than 1, considering Dehn twists on a very simple set of curves.

We show that every subgroup of the mapping class group MCG(S) of a compact surface S is either virtually abelian or it has infinite dimensional second bounded cohomology. As an application, we give another proof of the…

We establish that, given $\Sigma$ a compact orientable surface, and $G$ a finitely presented one-ended group, the set of copies of $G$ in the mapping class group $\mathcal{MCG}(\Sigma)$ consisting of only pseudo-anosov elements except…

This is an announcement of some of the results obtained as a part of the second author's Ph.D. thesis. In the first part, we prove that the fundamental group of an acylindrical complex of hyperbolic groups with finite edge groups is…

We investigate boundedness of hyperbolic components in the moduli space of Newton maps. For quartic maps, (i) we prove hyperbolic components possessing two distinct attracting cycles each of period at least two are bounded, and (ii) we…

This paper develops new tools for understanding surfaces with more than one end (and usually, of infinite topology) which properly minimally embed into Euclidean three-space. On such a surface, the set of ends forms a compact Hausdorff…

We introduce a new class of discrete conformal structures on surfaces with boundary, which have nice interpolations in 3-dimensional hyperbolic geometry. Then we prove the global rigidity of the new discrete conformal structures using…

We consider the oriented graph whose vertices are isomorphism classes of finitely generated groups, with an edge from G to H if, for some generating set T in H and some sequence of generating sets S_i in G, the marked balls of radius i in…

We prove necessary and sufficient conditions on a family of (generalised) gridding matrices to determine when the corresponding permutation classes are partially well-ordered. One direction requires an application of Higman's Theorem and…

We introduce and geometrically characterize the notion of uniformly perfect Morse boundary for proper geodesic metric spaces. As a unifying result, we prove that the Morse boundary of any finitely generated, non-elementary group is…

In this paper, we classify homomorphisms from the braid group of $n$ strands to the mapping class group of a genus $g$ surface. In particular, we show that when $g<n-2$, all representations are either cyclic or standard. Our result is sharp…

In 1980's, Thurston established a combinatorial characterization for post-critically finite rational maps. This criterion was then extended by Cui, Jiang, and Sullivan to sub-hyperbolic rational maps. The goal of this paper is to present a…