Lvzhou(Joe) Chen 陈绿洲

About

I am an R. H. Bing postdoctoral fellow in the Department of Mathematics at UT Austin. I received my PhD from University of Chicago under the supervision of Danny Calegari.

I am interested in geometric topology, geometric group theory and dynamics. Topics I specifically studied include stable commutator length, which is a relative version of the Gromov-Thurston norm and dual to quasimorphisms, and mapping class groups of infinite-type surfaces (such as the plane minus a Cantor set, which occurs naturally in dynamics).

Here is my CV.

The letter "v" in my first name really stands for "ü", a missing vow in English, pronounced as in German. Simply call me Joe unless you are curious about the accurate pronunciation.

Photo

Photo by Le Zhuang


Publications

8. (with Danny Calegari) Normal subgroups of big mapping class groups, in preparation.

7. (with Alexander J. Rasmussen) Laminations and 2-filling rays on infinite type surfaces, preprint, 41 pages, arXiv 2010.06029.
Short description: We construct the so-called 2-filling rays on surfaces of infinite type, answering a question of Bavard-Walker. These rays are related to the understanding of the Gromov boundary of the loop graph of the surface in terms of geodesic laminations. We give a hands-on contruction as well as one using train tracks and laminations. The construction is robust and produces a continuum of 2-filling rays in distinct mapping class group orbits.

6. (with Santana Afton, Danny Calegari, Rylee Alanza Lyman) Nielsen realization for infinite-type surfaces, Proc. Amer. Math. Soc., to appear, 8 pages, arXiv 2002.09760.
Short description: For any orientable surface S of infinite type, we show any finite subgroup of its mapping class group can be realized as a group of isometries for some hyperbolic metric on S. We use this to classify torsion elements in certain big mapping class groups, and also show that topological groups containing sequences of torsion elements limiting to the identity do not embed continuously into the mapping class group of S. Finally, we show that compact subgroups of the mapping class group of S are finite, and locally compact subgroups are discrete.

5. (with Nicolaus Heuer) Spectral gap of scl in graphs of groups and 3-manifolds, submitted, 69 pages, arXiv 1910.14146.
Short description: We show a sharp lower bound 1/2 for the stable commutator length (scl) of any hyperbolic element in a group acting on a tree, provided that every edge stabilizer is relatively torsion-free in vertex stabilizers. The sharp bound becomes 1/2-1/n if the edge stabilizers are n-relatively torsion-free in vertex stabilizers. As an application, we show any element in a graph product of torsion-free groups has scl at least 1/2 unless it is supported in a complete subgraph. This in particular implies a sharp gap 1/2 for all right-angled Artin groups. We also characterize scl in vertex stabilizers, which allows us to show that any 3-manifold group has a spectral gap, that is a constant C such that any element g has either scl(g)>=C or scl(g)=0. We also classify elements whose scl vanish. The method is topological in nature. In many cases we also construct explicit quasimorphisms detecting these lower bounds, including a new short proof of the gap 1/2 in free groups.

4. (with Danny Calegari) Big mapping class groups and rigidity of the simple circle, Ergodic Theory and Dynamical Systems, to appear, 28 pages, arXiv 1907.07903.
Short description: Let G denote the mapping class group of the plane minus a Cantor set. We show that every action of G on the circle is either trivial or semi-conjugate to a unique minimal faithful action on the so-called simple circle. In the appendix, we compute the second homology of the mapping class group of the sphere minus a Cantor set.

3. Scl in graphs of groups, Invent. Math., 221, no. 2, 329-396.
Short description: Let G be a group acting on a tree with cyclic edge and vertex stabilizers. Then stable commutator length (scl) is rational in G. A large portion of the paper concentrates on the special case of Baumslag-Solitar groups, giving a criterion for existence of extremal surfaces, and proving that scl in so-call "surgery" families varies predictably and converges to scl in free groups. This is a homological analog of the phenomenon of geometric convergence in hyperbolic Dehn surgery.

2. Spectral gap of scl in free products, Proc. Amer. Math. Soc. 146 (2018), no.7, 3143-3151.
Short description: We show that for a free product of torsion-free groups, scl(g) is at least 1/2 for any g not conjugate into any factor. Similar sharp bounds are obtained assuming the factors have no small torsion. The method is topological and new for scl estimates, and is pushed further to groups acting on trees in "Spectral gap of scl in graphs of groups and applications" with Nicolaus Heuer. The method turns out to be closely related to Anton Klyachko's car motion argument used in his elegant proof of the Kervaire conjecture for free products of torsion-free groups.

1. Scl in free products, Algebr. Geom. Topol., 18 (2018), no. 6, 3279-3313.
Short description: We investigate scl in free products, and show that scl is rational in free products of groups with vanishing scl. We further prove that the property of isometric embedding with respect to scl is preserved under taking free products. The method of proof gives a way to compute scl in free products which lets us generalize and derive in a new way several well-known formulas. Finally we show that scl in free products of cyclic groups behaves in a piecewise quasi-rational way when the word is fixed but the orders of factors vary.

Notes

Topic Proposal: Stable Commutator Length

Slides

Big mapping class groups and rigidity of the simple circle, Hyperbolic Lunch at UToronto (virtual)
Stable commutator length in graphs of groups, NCNGT 2020
Spectral gap of scl, 2017 Fall AMS sectional meeting at Buffalo

Contact

Email: lvzhou.chen (followed by @math.utexas.edu)
Mail: Department of Mathematics
The University of Texas at Austin
2515 Speedway, PMA 8.100
Austin, TX 78712

Links

Research Blog of Danny Calegari
Low Dimensional Topology Blog
Course Blog on Geometric Group Theory by Henry Wilton