Related papers: Copies of a one-ended group in a Mapping Class Gro…

We prove that infinite mapping class group orbits are dense in the character variety of Deroin-Tholozan representations. In other words, the action is minimal except for finite orbits. Our arguments rely on the symplectic structure of the…

We prove that if $X = X_1 \times \dots \times X_n$ is a product of hyperbolic Riemann surfaces of finite type and $Y = \Omega/\Gamma$ is a complex manifold, where $\Omega$ is a bounded simply-connected domain in $\mathbb{C}^m$, then the…

Let S be an orientable surface of finite type. Using Pho-On's infinite unicorn paths, we prove the hyperfiniteness of orbit equivalence relations induced by the actions of the mapping class group of S on the Gromov boundaries of the arc…

Frucht showed that, for any finite group $G$, there exists a cubic graph such that its automorphism group is isomorphic to $G$. For groups generated by two elements we simplify his construction to a graph with fewer nodes. In the general…

We introduce moment maps for continuous unitary representations of general topological groups. For solvable separable locally compact groups, we prove that the closure of the image of the moment map of any representation is convex.

Semistability at infinity is an asymptotic property of finitely presented groups that is needed in order to effectively define the fundamental group at infinity for a 1-ended group. It is an open problem whether or not all finitely…

We describe the mapping class group action on the cohomology of the twisted $\mathrm{SL}_n$-character variety of a surface $\Sigma_g$ of genus $g$. Our main tool is a relative version of the endoscopic decomposition of Maulik-Shen; this…

In this work we consider foliations of compact manifolds whose holonomy pseudo-group is expansive, and analyze their number of compact leaves. Our main result is that in the codimension-one case this number is at most finite, and we give…

We show that for any $k$ at least $6$ and $g$ sufficiently large, the mapping class group of a surface of genus $g$ can be generated by three elements of order $k$. We also show that this can be done with four elements of order $5$. We…

Let R be a compact, connected, orientable surface of genus g with p boundary components. Let C(R) be the complex of curves on R and Mod_R^* be the extended mapping class group of R. Suppose that either g = 2 and p > 1 or g > 2 and p >= 0.…

For a genus g handlebody H a simplicial complex, with vertices being isotopy classes of certain incompressible surfaces in H, is constructed and several properties are established. In particular, this complex naturally contains, as a…

We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides…

We prove that there are only finitely many conjugacy classes of arithmetic maximal hyperbolic reflection groups.

We present a uniform methodology for computing with finitely generated matrix groups over any infinite field. As one application, we completely solve the problem of deciding finiteness in this class of groups. We also present an algorithm…

We show that for any closed surface of genus greater than one and for any finite weighted graph filling the surface, there exists a hyperbolic metric which realizes the least Dirichlet energy harmonic embedding of the graph among a fixed…

Let $\Sigma$ be a compact surface. We prove that the set of surface cubications modulo flips, up to isotopy, is in one-to-one correspondence with $\Z/2\Z\oplus H_1(\Sigma,\Z/2\Z)$.

We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of…

We show that any isomorphism between mapping class groups of orientable infinite-type surfaces is induced by a homeomorphism between the surfaces. Our argument additionally applies to automorphisms between finite-index subgroups of these…

Let Gamma_k be the lower central series of a surface group Gamma of a compact surface S with one boundary component. A simple question to ponder is whether a mapping class of S can be determined to be pseudo-Anosov given only the data of…

Let $G$ be a definable group in a $p$-adically closed field $M$. We show that $G$ has finitely satisfiable generics (fsg) if and only if $G$ is definably compact. The case $M = \mathbb{Q}_p$ was previously proved by Onshuus and Pillay.