arXiv subjects
Publications and source records attributed to Yanlong Hao.
For every closed surface $\Sigma_g$ of genus $g\geq6$, we prove that there exists a countable union of positive-codimension algebraic subsets of Teichm\"uller space such that, for every hyperbolic metric $d$ outside this exceptional set, the number of distinct primitive closed-geodesic lengths at most $L$ is bounded below by $\tau(d)e^{\delta(g)L}$, where the explicit constant $\delta(g)$ satisfies $\delta(g)>\frac12$.
We study subgroups generated by prescribed unipotent elements. For $n\geq 3$, let \[ \Gamma(Q)=\langle E_{ij}(q_{ij}):i\neq j\rangle \] be the subgroup of $\operatorname{SL}(n,\mathbb R)$ generated by elementary matrices with nonzero rational parameters $q_{ij}$. We prove that $\Gamma(Q)$ is always $S$-arithmetic, extending classical integral-parameter results to arbitrary rational parameters. Our method is effective: it determines the relevant ring of $S$-integers, a diagonal conjugating matrix, and an explicit description of the resulting subgroup by congruence conditions. We then study the rank-one family \[ \Gamma_q= \left\langle \begin{pmatrix} 1&1\\ 0&1 \end{pmatrix}, \begin{pmatrix} 1&0\\ q&1 \end{pmatrix} \right\rangle, \qquad q=\tfrac{s}{t}\in\mathbb Q. \] For $q\neq0,\pm3$, we prove that \[ \Gamma_q=\Gamma_1^{(t)}(s) \] if and only if its upper-triangular subgroup strictly contains \[\left\langle\begin{pmatrix}1&1\\0&1\end{pmatrix}\right\rangle.\] Thus the congruence-subgroup problem is reduced to constructing a single upper-triangular element outside this cyclic subgroup. As applications, we reinterpret several constructions from the study of non-freeness as constructions of arithmetic groups. We verify the criterion for all rational parameters $q=\tfrac{s}{t}\in(-4,4)$ with $1\leq |s|\leq21$, and obtain new infinite families of congruence subgroups from indefinite binary quadratic forms and Pell-type equations.
In this paper, we investigate the relationship between the divergence of Kleinian groups $\Gamma$ and the recurrence of simple random walks on the Schreier graph associated with $\Gamma$. In particular, we show that if $\Gamma$ is a subgroup of a lattice and is of divergence type, then the Schreier graph is recurrent. Our approach builds connections among the growth rate of the $\Gamma$-orbit, the volume growth rate of the quotient manifolds, and the growth rate of the Schreier graph. Using the connections, we construct abundant Kleinian groups of divergence type.
This paper gives an example of a non-arithmetic surface with marked length variety rigidity.
This paper aims to characterize rank-one arithmetic and locally symmetric metrics in the coarsely geometric setting using coarse-geometric commensurators. We provide a positive answer in general under the Hilbert-Smith conjecture and unconditionally for finite volume negatively curved manifolds with finitely many cusps.
In this paper, we investigate the trace set of a Fuchsian lattice. There are two results of this paper: the first is that for a non-uniform lattice, we prove Scmutz's conjecture: the trace set of a Fuchsian lattice exhibits linear growth if and only if the lattice is arithmetic. Additionally, we show that for a fixed surface group of genus bigger than 2 and any positive number $\epsilon$, te set of cocompact lattice embedding such that their growth rate of trace set exceeds $n^{2-\epsilon}$ has positive Weil-Petersson volume. We also provide an asymptotic analysis of the volume of this set.
Let $(X,d)$, $(Y, d')$ be two roughly geodesically complete Gromov hyperbolic spaces under comparable isometric actions of $Γ$. Assume that the limit set $ΛΓ=\partial X\partial Y$. If spaces $X$ and $Y$ have the same asymptotic marked length spectrum, meaning that $$\lim_{{l_{d}([γ])\to \infty}}\frac{l_d(γ)}{l_{d'}(γ)}=1.$$ Then $(X,d)$ and $(Y,d')$ are $Γ$-equivariantly roughly isometric.
In this short note, we show that the B-C property characterizes arithmetic lattices along all nonuniform lattices in ${\mathrm{PSL}}(2,\mathbb{C})$.
In this paper, we show that if a group acts isometrically on a good hyperbolic space of finite volume entropy through a non-elementary action, then it admits an affine action on some $L^p$ -space with an unbounded orbit for sufficiently large $p$. As an application, we prove that any isometric action of a group with the fixed point property $F_\infty$ on a good hyperbolic space must have a bounded orbit.
Let $\rm{Aut}(p)$ denote the space of all self-fibre homotopy equivalences of a principal $G$-bundle $p: E\rightarrow X$ of simply connected CW complexes with $E$ finite. When $G$ is a compact connected topological group, we show that there exists an inequality $$n-{\rm N}(p)\leq {\rm Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_0)\leq n$$ for any space $X$, where $n$ is the number of non-trivial rational homotopy groups of $G$ and ${\rm N}(p)$ is defined in Section 2. In particular, ${\rm Hnil}_{\mathbb{Q}}({\rm{Aut}}(p)_{0})=n$ if $p$ is a fibre-homotopy trivial bundle and X is finite.
In this paper, we describe the homotopy type of the homotopy fixed point sets of $S^3$-actions on rational spheres and complex projective spaces, and provide some properties of $S^1$-actions on a general rational complex.
In this note, we give some new families of two-stage spaces for which the torus rank conjecture is affirmed.
In this paper, we prove a cocycle version of marked length spectrum rigidity. There are two consequences. The first is marked length pattern rigidity for arithmetic hyperbolic locally symmetric manifolds. The second is strengthen marked length spectrum rigidity for surfaces and closed locally symmetric manifolds.
Moore's Conjecture is shown to hold for generalized moment-angle complexes and a criterion is proved that determines when a polyhedral product is elliptic or hyperbolic.
In this paper, we focus on some models in rational homotopy theory, Sullivan model, Quillen model, C_\infty model, and L_\infty model. We give some connections between them. As an application, we prove the Torus Rank Conjecture.