Thomas Ng - Research

Research

See below for abstracts.

5. On the boundary criterion for relative cubulation
with Eduard Einstein and Suraj Krishna MS-- (pdf)(arXiv)(journal)
Bulletin of the London Mathematical Society (2025)

4. Separation and relative quasi-convexity criteria for relatively geometric actions
with Eduard Einstein and Daniel Groves -- (pdf)(arXiv)(journal)
Groups, Geometry, and Dynamics (2024)

3. Hierarchically hyperbolic groups and uniform exponential growth
with Carolyn Abbott and Davide Spriano -- (pdf)(arXiv)(journal)
Appendix by Radhika Gupta and Harry Petyt
Mathematische Zeitschrift (2023)

2. Groups acting on CAT(0) cube complexes with uniform exponential growth
with Radhika Gupta and Kasia Jankiewicz -- (pdf)(arXiv)(journal)
Algebraic & Geometric Topology (2023)

1. Conformally covariant operators and conformal invariants on weighted graphs

with Dmitri Jakobson, Matthew Stevenson, and Mashbat Suzuki -- (pdf)(arXiv)(journal)
Geometriae Dedicata (2014)

Preprints:

8. Random quotients of free products

with Eduard Einstein, Suraj Krishna MS, MuphyKate Montee, and Markus Steenbock-- (pdf)(arXiv)(submitted)
2024

7. Relative cubulation of small cancellation free products

with Eduard Einstein -- (pdf)(arXiv)(submitted)
2021

6. Extensions of hyperbolic groups have locally uniform exponential growth

with Robert Kropholler and Rylee Lyman -- (pdf)(arXiv)(submitted)
2020

Recordings and slides from selected presentations:

"What is an HHG?" (Recording) from HHG Quotients Conference at (U. of Bristol 2022)

"Residually finite quotients of free products" (25 min) from SumTopo2022 Conference (U. Vienna 2022)

"Uniform exponential growth in automorphism groups" (5 min) from MCG and OutFn Conference (IHP 2022)

Slides (6 min) at Counting problems (Ventotene 2021)

Slides (5 min) at Aspects of non-positive and negative curvature in group theory (CIRM 2019).

Overview:

My research addresses uniform exponential growth of groups with various notions of nonpositive curvature including CAT(0), relative hyperbolicity, acylindrical hyperbolicity, hierarchical hyperbolicity.

Broad research interests: Geometric group theory. Low-dimensional topology. Combinatorics. Formal languages.

Favorite objects of study: (Right-angled) Artin/Coxeter groups. CAT(0) cube complexes. Surfaces and associated cell complexes: arc, curve, pants. Free-by-cyclic groups. Hyperbolic 3-manifolds. Mapping class groups. Artin groups.

Tools I enjoy using: Nonpositive curvature. Classification of isometries. Quasiconvexity. Bass-Serre theory. Acylindricity. Complexes of groups. Ramsey theory. Automatic structures on groups: regular languages of geodesics.

Detailed descriptions (abstracts) of papers:

On the boundary criterion for relative cubulation8. Random quotients of free products

with Eduard Einstein, Suraj Krishna MS, MuphyKate Montee, and Markus Steenbock-- (pdf)
Abstract: We introduce a density model for random quotients of a free product of finitely generated groups. We prove that a random quotient in this model has the following properties with overwhelming probability: if the density is below 1/2, the free factors embed into the random quotient and the random quotient is hyperbolic relative to the free factors. Further, there is a phase transition at 1/2, with the random quotient being a finite group above this density. If the density is below 1/6, the random quotient is cubulated relative to the free factors. Moreover, if the free factors are cubulated, then so is the random quotient.

On the boundary criterion for relative cubulation

with Eduard Einstein and Suraj Krishna MS-- (pdf)
Abstract: In this note, we clarify that the boundary criterion for relative cubulation of the first author and Groves works even when the peripheral subgroups are not one-ended. Specifically, if the boundary criterion is satisfied for a relatively hyperbolic group, we show that, up to taking a refined peripheral structure, the group admits a relatively geometric action on a CAT(0) cube complex. We anticipate that this refinement will be useful for constructing new relative cubulations in a variety of settings.

Relative cubulation of small cancellation free products

with Eduard Einstein -- (pdf)
Abstract: We expand the class of groups with relatively geometric actions on CAT(0) cube complexes by proving that it is closed under C'(1/6)--small cancellation free products. We build upon a result of Martin and Steenbock who prove an analogous result in the more specialized setting of groups acting properly and cocompactly on CAT(0) cube complexes. Our methods make use of the same blown-up complex of groups to construct a candidate collection of walls. However, rather than arguing geometrically, we show relative cubulation by appealing to a boundary separation criterion of Einstein and Groves and proving that wall stabilizers form a sufficiently rich family of full relatively quasi-convex codimension-one subgroups. The additional flexibility of relatively geometric actions has surprising new applications. For example, we prove that C'(1/6)--small cancellation free products of residually finite groups are residually finite.

Separation and relative quasi-convexity criteria for relatively geometric actions

with Eduard Einstein and Daniel Groves -- (pdf)(arXiv)
Abstract: Bowditch characterized relative hyperbolicity in terms of group actions on fine hyperbolic graphs with finitely many edge orbits and finite edge stabilizers. In this paper, we define generalized fine actions on hyperbolic graphs, in which the peripheral subgroups are allowed to stabilize finite sub-graphs rather than stabilizing a point. Generalized fine actions are useful for studying groups that act relatively geometrically on a CAT(0) cube complex, which were recently defined by the first two authors. Specifically, we show that a group acting relatively geometrically on a CAT(0) cube complex admits a generalized fine action on the one-skeleton of the cube complex. For generalized fine actions, we prove a criterion for relative quasi-convexity as subgroups that cocompactly stabilize quasi-convex sub-graphs, generalizing a result of Martinez-Pedroza and Wise in the setting of fine hyperbolic graphs. As an application, we obtain a characterization of boundary separation in generalized fine graphs and use it to prove that Bowditch boundary points in relatively geometric actions are always separated by a hyperplane stabilizer.

Extensions of hyperbolic groups have locally uniform exponential growth

with Robert Kropholler and Rylee Lyman -- (pdf)(arXiv)
Abstract: We introduce a quantitative characterization of subgroup alternatives modeled on the Tits alternative in terms of group laws and investigate when this property is preserved under extensions. We develop a framework that lets us expand the classes of groups known to have locally uniform exponential growth to include extensions of either word hyperbolic or right-angled Artin groups by groups with locally uniform exponential growth. From this, we deduce that the automorphism group of a torsion-free one-ended hyperbolic group has locally uniform exponential growth. Our methods also demonstrate that automorphism groups of torsion-free one-ended toral relatively hyperbolic groups and certain right-angled Artin groups satisfy our quantitative subgroup alternative.
(Presentation notes)

Groups acting on CAT(0) cube complexes with uniform exponential growth

with Radhika Gupta and Kasia Jankiewicz -- (pdf)(arXix)
Abstract: We study uniform exponential growth of groups acting properly on CAT(0) cube complexes in terms of dimension of the cube complex. Work in this context was initiated by Kar and Sageev in the setting of torsion-free groups acting on 2-dimensional cube complexes. We extend their results to any group acting without global fixed point on a 2-dimensional cube complex, by producing uniformly short loxodromic isometries from elliptic one. For torsion-free groups, we show that under non-virtually abelian groups that act on cube complexes with isolated flats have uniform exponential growth depending only on the cubical dimension.
The slides I used for a five-minute lightning talk at Redbud conference in April 2019.A poster I presented on at YGGT8 in July 2019

Hierarchically hyperbolic groups and uniform exponential growth

with Carolyn Abbott and Davide Spriano -- (pdf)(arXiv)
Abstract: We give several sufficient conditions for uniform exponential growth in the setting of virtually torsion-free hierarchically hyperbolic groups that apply to most known examples. For example, any hierarchically hyperbolic group that is also acylindrically hyperbolic has uniform exponential growth depending only on constants associated to the geometry of the hierarchical structure.In addition, we characterize hierarchically hyperbolic groups without uniform exponential growth as those quasi-isometric to a product, one of whose factors is a line, and give restrictions on how such a group may act on the collection of domains. This work expands upon past results on the coarse geometry of hierarchically hyperbolic groups and provides several new structural results both on the shape of a hierarchy as well as a more effective classification of the behavior of collections of infinite order elements. Our methods give a new unified proof of uniform exponential growth for several examples of groups with notions of non-positive curvature and under additional hypotheses provide a quantitative Tits alternative for hierarchically hyperbolic groups. In particular, we obtain the first proof of uniform exponential growth for certain groups that act geometrically on CAT(0) cubical groups of dimension 3 or more.

Conformally covariant operators and conformal invariants on weighted graphs

with Dmitri Jakobson, Matthew Stevenson, and Mashbat Suzuki -- (pdf) (arXiv)
At a glance: We consider weighted finite connected simple graphs equipped. We give a notion of conformal equivalence of graphs weights and describe the moduli space all such conformal structures. Based on this, we propose a definition of conformally covariant operators on graphs motivated by work of Graham, Jenne, Mason, and Sparling, and provide several examples of such operators including the adjacency matrix and edge Laplacian of the graph. In the setting of when such an operator has nontrivial kernel, we construct conformal invariants of graphs in direct analogy to several results of Canzani, Grover, Jakobson, and Ponge established for Riemannian manifolds. In particular, we show that the nodal sets and nodal domains of null eigenvectors are conformal invariants.
A poster highlighting main results.
A short expository article describing the analogy between the spectral theory of weighted graphs with that of Reimannian manifolds.

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