In this talk, we will see a few approaches to obtain surfaces with either hyperbolic structures or singular Euclidean structures from 1-dimensional dynamical systems. A part of the project is related to the Fried's conjectural characterization of pseudo-Anosov stretching factors. If time permits, we will also briefly mention a statistical answer to an easy version of Fried's conjecture. This talk is partially based on joint work with J. Alonso-E. Samperton and A. Rafiqi-C. Wu.
We study the smallest positive eigenvalue of the Laplace-Beltrami operator on a closed hyperbolic 3-manifold which fibers over the circle. Using so-called Lipschitz model developed by Minsky and Brock-Canary-Minsky, we find a family of graphs which are uniformly quasi-isometric to such 3-manifolds. This implies that the smallest positive eigenvalue on such a graph and a manifold are uniformly comparable. Using this idea, we compute the eigenvalue on such graphs, and obtain essentially sharp upper bound. This is a joint-work with I. Gekhtman and U. Hamenstaedt.
TBA
In this talk we shall prove smooth rigidity on aspherical locally homogeneous manifolds as a final version. This result was partially announced in 2012. Let $G$ be a connected group and H a maximal compact subgroup such that $G/H$ is a divisible contractible manifold. That is, there exists a discrete cocompact subgroup $\pi$ of $G$. Then $\pi$ acts properly discontinuously on $G/H$ on the left. The compact quotient $M = \pi\backslash G/H$ is said to be an aspherical locally homogeneous manifold. Let $G = R\cdot S$ be a Levi-decomposition. Assume that the center of $S$ is finite and without $\mathrm{SL}(2, \mathbb{R})$-factor.
Theorem 1. Let $\varphi : \pi_1(M) \to \pi_1(M')$ be an isomorphism between the fundamental groups of compact aspherical locally homogeneous manifolds $M$ and $M'$. Then $\varphi$ is induced by a diffeomorphism $\Psi : M\to M'$.
This is a joint work with Oliver Baues (IIMT University of Fribourg).
The notion of Anosov representation is a generalization of convex cocompact representation in hyperbolic 3-space to higher rank symmetric spaces. In this talk we shall introduce this notion by looking at typical examples in the $\mathbf{PGL}(3, \mathbb{R})$ character variety. For example, hyperconvex representations introduced by Labourie and horocyclic representations introduced by Barbot when the domain group is the surface group, and positive representations introduced by Fock and Goncharov, and Pappus representations introduced by Schwartz when the domain group is the free group (of rank 2). While discussing these examples, we shall focus on the property of boundary-embeddedness (slightly weaker notion than Anosov) with our motivational questions in mind for the second talk. (Work in progress joint with Jaejeong Lee)
As shown in the first talk, among Anosov representations in the $\mathbf{PGL}(3, \mathbb{R})$ character variety of the free group, there are represen- tations that are negative in a certain sense, as well as the well-known positive (hyper-convex) representations. In this second talk, we focus on such representations which come from the framed representations (in the sense of Fock-Goncharov) of the 3-holed sphere group. In order to exhibit concrete examples we restrict our attention to the (real 2-dimensional) relative character variety where the bound- ary monordomies are all quasi-unipotent. Boundary-embedded and transversal (or antipodal) representations in this subvariety may be called relatively Anosov and all such examples properly include the Pappus representations studied by R. Schwartz in 1993. We further restrict to a certain real 1-dimensional subvariety con- sisting of representations with 2-fold symmetry and find all boundary- embedded ones in there, thereby obtaining a $\mathbf{PGL}(3,\mathbb{R})$ analogue of R. Schwartz’s solution to the Goldman-Parker conjecture on the ideal triangle reflection groups in $\mathbf{PU}(2, 1)$. (Work in progress joint with Sungwoon Kim)
Let $M$ be a hyperbolic fibered 3-manifold with $b_1(M) \geq 2$ and let $S$ be a fiber with pseudo-Anosov monodromy $\psi$. We show that there exists a sequence $(R_n, \psi_n)$ of fibers contained in the fibered cone of $(S,\psi)$ such that the asymptotic translation length of $\psi_n$ on the curve complex $\mathcal{C}(R_n)$ behaves asymptotically like $1/|\chi(R_n)|^2$. As an application, we can reprove the previous result by Gadre--Tsai that the minimal asymptotic translation lengths of a closed surface of genus $g$ are bounded below and above by $C/g^2$ and $D/g^2$ for some positive constants $C$ and $D$, respectively. We also show that this also holds for the cases of hyperelliptic mapping class group and hyperelliptic handlebody group. This is a joint work with Eiko Kin.
Let $f:M_0\to M_1$ be a degree-one map between closed oriented hyperbolic manifolds of dimension $\geq 3$. Thurston proved by using results of Gromov's works that $f$ is homotopic to an isometry if and only if $\mathrm{vol}(M)=\mathrm{vol}(N)$. This theorem suggests us Gromov-Thurston's principle on hyperbolic geometry of dimension three (or more) that "Volume determines the structure". The aim of my talk is to present an infinite volume version of the principle.
Swapping algebra is a Poisson algebra defined on the polynomial ring given by pairs of points on a circle. In my thesis, I study the Poisson quotient of the swapping algebra. It induces a Poisson algebra on a geometric invariant ring. F. Labourie assicated the swapping algebra to Goldman symplectic structure and this space of real opers with trivial holonomy. In my recent work, I associated the rank n swapping algebra to Fock--Goncharov Poisson algebra on the Fock--Gocharov $X$ space. Fock--Goncharov Poisson algebra is a canonical Poisson structure on the assocaited quiver representation for the cluster algebraic structure. We will show that the cluster dynamic can be characterized by the Plucker relation on the rank $n$ swapping algebra. Also for thier quantized version. If time premitted. I will also show how to work on the discrete integrable system by using rank $n$ swapping algebra.
The group $Out(F_2)$ of outer-automorphisms of $F_2$ the rank 2 free group acts naturally on the $PSL(2,\mathbb{C})$-character variety of $F_2$. To study the dynamical property of $Out(F_2)$-action, Bowditch's $Q$-condition and the primitive stable condition on a representation from $F_2$ to $PSL(2,\mathbb{C})$ have been introduced by Bowditch (generalized by Tan-Wong-Zhang) and by Minsky, respectively. Each one of them can characterize an open subset of the character variety on which $Out(F_2)$ acts properly discontinuously. These two open sets are both candidates for the maximal domain of discontinuity for the $Out(F_2)$-action. In a joint work with Jaejeong Lee, we show that these two conditions are equivalent to each other.