Related papers: Subgroups generated by two pseudo-Anosov elements …

Let $M =\mathbb{H}^3/\Gamma$ be a finite-volume, noncompact hyperbolic 3-manifold. We show that the number of quasi-Fuchsian surface subgroups of $\Gamma$ (up to conjugacy and commensurability) of genus at most $g$ is bounded both above and…

We count and give a parametrization of connected components in the space of flags transverse to a given transverse pair in every flag varieties of $\mathrm{SO}_0(p,q)$. We compute the effect the involution of the unipotent radical has on…

We prove that Anosov representations from a surface group to SL(3,R) are uniquely determined by their boundary maps if and only if they do not factor over a completely reducible representation. Furthermore we discuss representations not…

Let $G$ be a connected semisimple real algebraic group. We prove that limit cones vary continuously under deformations of Anosov subgroups of $G$ under a certain convexity assumption, which turns out to be necessary. We apply this result to…

Let $M(N_{h,n})$ denote the mapping class group of a compact nonorientable surface of genus $h\ge 7$ and $n\le 1$ boundary components, and let $T(N_{h,n})$ be the subgroup of $M(N_{h,n})$ generated by all Dehn twists. It is known that…

Fix a finite group $G$. We study $\Omega^{SO,G}_2$ and $\Omega^{U,G}_2$, the unitary and oriented bordism groups of smooth $G$-equivariant compact surfaces, respectively, and we calculate them explicitly. Their ranks are determined by the…

Let S be a surface and let Mod(S,K) be the mapping class group of S permuting a Cantor subset K of S. We prove two structure theorems for normal subgroups of Mod(S,K). (Purity:) if S has finite type, every normal subgroup of Mod(S,K) either…

McCarthy's Theorem for the mapping class group of a closed hyperbolic surface states that for any two mapping classes $\sigma,\tau \in \mathrm{Mod}(S)$ there is some power $N$ such that the group $\langle \sigma^N,\tau^N\rangle$ is either…

Let $BS(1,n)= <a,b : a b a ^{-1} = b ^n>$ be the solvable Baumslag-Solitar group, where $n \geq 2$. We study representations of $BS(1, n)$ by homeomorphisms of closed surfaces with (pseudo-)Anosov elements. That is, we consider a closed…

We prove two results. (1) There is an absolute constant $D$ such that for any finite quasisimple group $S$, given 2D arbitrary automorphisms of $S$, every element of $S$ is equal to a product of $D$ `twisted commutators' defined by the…

In this paper, we make use of the relations between the braid and mapping class groups of a compact, connected, non-orientable surface N without boundary and those of its orientable double covering S to study embeddings of these groups and…

There is a forgetful map from the mapping class group of a punctured surface to that of the surface with one fewer puncture. We prove that finitely generated purely pseudo-Anosov subgroups of the kernel of this map are convex cocompact in…

We know from previous work with Italiano and Migliorini that there exists some hyperbolic 5-manifold that fibers over the circle. Here we build one example where the monodromy is a "pseudo-Anosov homeomorphism" of the 4-dimensional fiber,…

Given a finite generating set $S$, let us endow the mapping class group of a closed hyperbolic surface with the word metric for $S$. We discuss the following question: does the proportion of non-pseudo-Anosov mapping classes in the ball of…

A conjecture of Andersen, Masbaum and Ueno states that for any compact oriented surface $\Sigma_{g,n}$ and any pseudo-Anosov $f\in \mathrm{Mod}(\Sigma_{g,n}),$ the matrix $\rho_r(f)$ has infinite order for any large $r,$ where $\rho_r$ is…

We show that certain classes of graphs of free groups contain surface subgroups, including groups with positive $b_2$ obtained by doubling free groups along collections of subgroups, and groups obtained by "random" ascending HNN extensions…

Given a finite dimensional manifold $N$, the group $\operatorname{Diff}_{\mathcal S}(N)$ of diffeomorphism of $N$ which fall suitably rapidly to the identity, acts on the manifold $B(M,N)$ of submanifolds on $N$ of diffeomorphism type $M$…

We compute the growth series and the growth functions of reducible and pseudo-Anosov elements of the pure mapping class group of the sphere with four holes with respect to a certain generating set. We prove that the ratio of the number of…

We determine which closed orientable $3$-manifolds $M$ admit a self-homeomorphism restricting to a pseudo-Anosov map on an incompressible subsurface $\Sigma$, which we call a pseudo-Anosov surface. When $M$ is irreducible, we show that the…

Suppose a and b are distinct isotopy classes of essential simple closed curves in an orientable surface S. Let T_a and T_b represent the respective Dehn twists along a and b. In this paper, we study the subgroups of Mod(S) generated by X…