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Sangrok Oh

Publications and source records attributed to Sangrok Oh.

8 recordsLinked to original sources
arXiv2026

Hierarchical geometry and right-angled Artin groups in graph braid groups

For the unordered discrete configuration space $\mathrm{UD}_n(\mathsf{\Gamma})$ of $n$ particles on a connected finite graph $\mathsf{\Gamma}$, we construct an explicit factor system on its universal cover. Its factors are encoded by legal pairs, namely subgraphs equipped with particle distributions. The nesting, orthogonality, and product regions in the resulting hierarchically hyperbolic group (HHG) structure admit explicit descriptions in terms of configuration-space geometry, and we show that this structure satisfies the additional properties needed for constructing and obstructing subgroups isomorphic to right-angled Artin groups (RAAGs). Using sufficiently subdivided models, we apply this hierarchy to graph braid groups. We give a finite combinatorial formula for the maximal rank of a free abelian subgroup and show that every RAAG occurs as an undistorted subgroup of some graph braid group. For graph $2$-braid groups, we obtain stronger restrictions: every RAAG subgroup has bipartite defining graph, and the embedding problem is characterized by an induced-subgraph condition in the expanded core graph of the hierarchy. For the RAAG defined by the four-vertex path, this condition is equivalent to a finite graphical criterion on the underlying graph.

math.GR↗
arXiv2026

From algebraic orthogonality to RAAG embedding obstructions in hierarchically hyperbolic groups

We introduce the $\textit{expanded core graph}$, which records minimal unbounded domains and their axial directions, for two classes of hierarchically hyperbolic group structures modeled on compact special groups and mapping class groups. Our main structural result shows that every embedding of a right-angled Artin group, after replacing its standard generators by positive powers, factors through an intermediate RAAG generated by suitably supported axial elements; for the class modeled on compact special groups, this intermediate RAAG is quasi-isometrically embedded. We show that its extension graph embeds into the expanded core graph. This yields a Kim--Koberda-type obstruction to RAAG embeddings and a complete embedding criterion when the rank is at most two. For the standard HHG structure on a mapping class group, the expanded core graph is the disjointness graph of essential curves, while for natural rich-family structures on a RAAG it recovers the extension graph. We also establish permanence results under finite direct products and the standard relatively hyperbolic construction.

math.GR↗
arXiv2026

On the large-scale geometry of graph braid groups via cubical structures

We study the large-scale geometry of graph braid groups $\mathbb{B}_n(\mathsf{\Gamma})$, viewed as the fundamental groups of discrete configuration spaces $UD_n(\mathsf{\Gamma})$, which are special cube complexes in the sense of Haglund--Wise. Exploiting this cubical structure, we relate hyperbolicity, undistorted surface subgroups, and group-theoretic decompositions. As a consequence, we obtain a complete classification of when $\mathbb{B}_n(\mathsf{\Gamma})$ is quasi-isometric to a free group via a purely geometric argument independent of discrete Morse theory. We then focus on graph $2$-braid groups. Using maximal product subcomplexes of $UD_2(\mathsf{\Gamma})$ and the intersection complex introduced in \cite{Oh22}, we show that, under natural assumptions, their union captures essential quasi-isometry information about $\mathbb{B}_2(\mathsf{\Gamma})$. As applications, we construct infinitely many graph $2$-braid groups that are quasi-isometric to right-angled Artin groups and infinitely many that are not, extending \cite{Oh22}, and we exhibit new phenomena in relative hyperbolicity.

math.GT↗
arXiv2025

Embeddability of right-angled Artin groups into hierarchically hyperbolic groups

For a hierarchically hyperbolic group, we give sufficient conditions ensuring that suitable powers of a finite collection of elements generate a right-angled Artin subgroup; under an additional condition, this subgroup is undistorted. We verify these hypotheses in two natural situations: one modeled on mapping class groups, using structural assumptions on the ambient HHG, and one modeled on RAAGs, requiring the chosen elements to be rigidly fully supported. Our results recover known embedding theorems for mapping class groups and extend the non-annular undistortion theorem of Clay--Leininger--Mangahas, while also recovering the Kim--Koberda extension graph theorem for RAAGs.

math.GR↗
arXiv2025

Quasi-isometry classification of certain graph $2$-braid groups and its applications

In \cite{Oh22}, the second author defined a complex of groups decomposition of the fundamental group of a finitely generated 2-dimensional special group, called an \emph{intersection complex}, which is a quasi-isometry invariant. In this paper, using the theory of intersection complexes, we classify the class of 2-braid groups over graphs with circumference $\leq 1$ up to quasi-isometry. Moreover, we find a sufficient condition when such a graph 2-braid group is quasi-isometric to a right-angled Artin group or not. Finally, by applying the same method, we also find that there is an algorithm to determine whether two 4-braid groups over trees are quasi-isometric or not.

math.GR↗
arXiv2022

Liftable automorphisms of right-angled Artin groups

Given a regular covering map $\varphi:\Lambda \to \Gamma$ of graphs, we investigate the subgroup $\operatorname{LAut}(\varphi)$ of the automorphism group $\operatorname{Aut}(A_\Gamma)$ of the right-angled Artin group $A_\Gamma$. This subgroup comprises all automorphisms that can be lifted to automorphisms of $A_\Lambda$. We first show that $\operatorname{LAut}(\varphi)$ is generated by a finite subset of Laurence's elementary automorphisms. For the subgroup $\operatorname{FAut}(\varphi)$ of $\operatorname{Aut}(A_\Lambda)$, which consists of lifts of automorphisms in $\operatorname{LAut}(\varphi)$, there exists a natural homomorphism $\operatorname{FAut}(\varphi)\to\operatorname{LAut}(\varphi)$ induced by $\varphi$. We then show that the kernel of this homomorphism is virtually a subgroup of the Torelli subgroup $\operatorname{IA}(A_\Lambda)$ and deduce a short exact sequence reminiscent of results from the Birman--Hilden theory for surfaces.

math.GR↗
arXiv2020

Quasi-isometry invariants of weakly special square complexes

We define the intersection complex for the universal cover of a compact weakly special square complex and show that it is a quasi-isometry invariant. By using this quasi-isometry invariant, we study the quasi-isometric classification of 2-dimensional right-angled Artin groups and planar graph 2-braid groups. Our results cover two well-known cases of 2-dimensional right-angled Artin groups: (1) those whose defining graphs are trees and (2) those whose outer automorphism groups are finite. Finally, we show that there are infinitely many graph 2-braid groups which are quasi-isometric to right-angled Artin groups and infinitely many which are not.

math.GR↗