arXiv subjects

Yusen Long

Publications and source records attributed to Yusen Long.

6 recordsLinked to original sources

arXiv2026

Fine projection complex and subsurface homeomorphisms with positive stable commutator length

Drawing inspiration from [BBF15], we construct a family of unbounded quasi-trees for a connected closed oriented surface $S_g$ of genus $g\geq 2$, upon which the group $\mathrm{Homeo}_0(S_g)$ acts coboundedly by isometries. As an application, we show that some surface homeomorphisms preserving a non-sporadic essential subsurface or an essential subsurface homeomorphic to a once-bordered torus can have positive stable commutator length in $\mathrm{Homeo}_0(S_g)$. Moreover, we provide a version of projection complex that does not require the finiteness condition.

math.GT↗

arXiv2025

An explicit exotic representation of a rank-one simple Lie group via convex bodies

In [DP12], Delzant and Py showed that there exist continuous irreducible isometric actions of $\mathrm{PSL}_2(\mathbb{R})$ on the infinite-dimensional hyperbolic space $\mathbb{H}^\infty$. Such continuous irreducible actions do not exist on the hyperbolic spaces $\mathbb{H}^n$ when $n>2$ and their associated embeddings $\mathbb{H}^2 \to \mathbb{H}^\infty$ given by the orbit maps were later called \emph{exotic} by Monod and Py in [MP14]. In this article, we produce a continuous and irreducible representation of $\mathrm{PSL}_2(\mathbb{R})\to \mathrm{Isom}(\mathbb{H}^\infty)$ using the hyperbolic model for convex bodies introduced in [DF22]. This yields a convex cocompact $\mathrm{PSL}_2(\mathbb{R})$-action on the infinite-dimensional hyperbolic space $\mathbb{H}^\infty$, of which the compact quotient over the minimal $\mathrm{PSL}_2(\mathbb{R})$-invariant convex set is homeomorphic to the 2-dimensional oriented Banach--Mazur compactum. Moreover, we study the geometry of one of its orbit maps and compute the Hausdorff dimension of the limit set of this representation.

math.GR↗

arXiv2025

Non-amenability of mapping class groups of infinite-type surfaces and graphs

This paper completely determines the non-amenability of the mapping class groups of infinite-type surfaces, the mapping class groups of locally finite infinite graphs of higher ranks, gives an example of non-amenable stabiliser of a point at infinity of a coarsely bounded generated hyperbolic Polish group, and exhibits a class of mapping class groups of trees or rank-one graphs that are amenable.

math.GR↗

arXiv2024

Connectedness of the Gromov boundary of fine curve graphs

The fine curve graph was introduced to study homeomorphism group of surfaces. In this paper we study the topology of the Gromov boundary of this graph for closed surfaces with higher genus. We first prove a bounded geodesic image theorem for the fine curve graph, a consequence of which is the non-compactness of the Gromov boundary. Using this theorem, we are able to show that the Gromov boundary is linearly connected with respect to some visual metric.

math.GT↗

arXiv2023

Big mapping class groups are not extremely amenable

This paper uses the renowned Kechris-Pestov-Todor\v{c}evi\'{c} machinery to show that (big) mapping class groups are not extremely amenable unless the underlying surface is a sphere or a once-punctured sphere, or equivalently when the mapping class group is trivial. The same techniques also show that the pure mapping class groups, as well as compactly supported mapping class groups, of a surface with genus at least one can never be extremely amenable.

math.GT↗

arXiv2023

Hyperbolic embedding of infinite-dimensional convex bodies

In this article, we use the second intrinsic volume to define a metric on the space of homothetic classes of Gaussian bounded convex bodies in a separable real Hilbert space. Using kernels of hyperbolic type, we can deduce that this space is isometrically embedded into an infinite-dimensional real hyperbolic space. Applying Malliavin calculus, it is possible to adapt integral geometry for convex bodies in infinite dimensions. Moreover, we give a new formula for computing second intrinsic volumes of convex bodies and offer a description of the completion for the hyperbolic embedding of Gaussian bounded convex bodies with dimension at least two and thus answer a question asked by Debin and Fillastre [DF22].

math.MG↗