arXiv subjects

Sang-hyun Kim

Publications and source records attributed to Sang-hyun Kim.

43 records · Page 3Linked to original sources

arXiv2012

An obstruction to embedding right-angled Artin groups in mapping class groups

For every orientable surface of finite negative Euler characteristic, we find a right-angled Artin group of cohomological dimension two which does not embed into the associated mapping class group. For a right-angled Artin group on a graph $\gam$ to embed into the mapping class group of a surface $S$, we show that the chromatic number of $\gam$ cannot exceed the chromatic number of the clique graph of the curve graph $\mathcal{C}(S)$. Thus, the chromatic number of $\gam$ is a global obstruction to embedding the right-angled Artin group $A(\gam)$ into the mapping class group $\Mod(S)$.

math.GT↗

arXiv2012

Surface subgroups of graph products of groups

A graph product kernel means the kernel of the natural surjection from a graph product to the corresponding direct product. We prove that a graph product kernel of countable groups is special, and a graph product of finite or cyclic groups is virtually cocompact special in the sense of Haglund and Wise. The proof of this yields conditions for a graph over which the graph product of arbitrary nontrivial groups (or some cyclic groups, or some finite groups) contains a hyperbolic surface group. In particular, the graph product of arbitrary nontrivial groups over a cycle of length at least five, or over its opposite graph, contains a hyperbolic surface group. For the case when the defining graphs have at most seven vertices, we completely characterize right-angled Coxeter groups with hyperbolic surface subgroups.

math.GR↗

arXiv2011

Embedability between right-angled Artin groups

In this article we study the right-angled Artin subgroups of a given right-angled Artin group. Starting with a graph $\gam$, we produce a new graph through a purely combinatorial procedure, and call it the extension graph $\gam^e$ of $\gam$. We produce a second graph $\gam^e_k$, the clique graph of $\gam^e$, by adding extra vertices for each complete subgraph of $\gam^e$. We prove that each finite induced subgraph $Λ$ of $\gam^e$ gives rise to an inclusion $A(Λ)\to A(\gam)$. Conversely, we show that if there is an inclusion $A(Λ)\to A(\gam)$ then $Λ$ is an induced subgraph of $\gam^e_k$. These results have a number of corollaries. Let $P_4$ denote the path on four vertices and let $C_n$ denote the cycle of length $n$. We prove that $A(P_4)$ embeds in $A(\gam)$ if and only if $P_4$ is an induced subgraph of $\gam$. We prove that if $F$ is any finite forest then $A(F)$ embeds in $A(P_4)$. We recover the first author's result on co--contraction of graphs and prove that if $\gam$ has no triangles and $A(\gam)$ contains a copy of $A(C_n)$ for some $n\geq 5$, then $\gam$ contains a copy of $C_m$ for some $5\le m\le n$. We also recover Kambites' Theorem, which asserts that if $A(C_4)$ embeds in $A(\gam)$ then $\gam$ contains an induced square. Finally, we determine precisely when there is an inclusion $A(C_m)\to A(C_n)$ and show that there is no "universal" two--dimensional right-angled Artin group.

math.GR↗

arXiv2010

Polygonal words in free groups

A longstanding question of Gromov asks whether every one-ended word-hyperbolic group contains a subgroup isomorphic to the fundamental group of a closed hyperbolic surface. An infinite family of word-hyperbolic groups can be obtained by taking doubles of free groups amalgamated along words that are not proper powers. We define a set of polygonal words in a free group of finite rank, and prove that polygonality of the amalgamating word guarantees that the associated square complex virtually contains a $π_1$-injective closed surface. We provide many concrete examples of classes of polygonal words. For instance, in the case when the rank is 2, we establish polygonality of words without an isolated generator, and also of almost all simple height 1 words, including Baumslag--Solitar relator $a^p (a^q)^b$ for $pq\ne0$.

math.GR↗

arXiv2010

Geometricity and Polygonality in Free Groups

Gordon and Wilton recently proved that the double D of a free group F amalgamated along a cyclic subgroup C of F contains a surface group if a generator w of C satisfies a certain 3-manifold theoretic condition, called virtually geometricity. Wilton and the author defined the polygonality of w which also guarantees the existence of a surface group in D. In this paper, virtual geometricity is shown to imply polygonality up to descending to a finite-index subgroup F' and applying an automorphism on F'. That the converse does not hold will follow from an example formerly considered by Manning.

math.GR↗

arXiv2009

On right-angled Artin groups without surface subgroups

We study the class N of graphs, the right-angled Artin groups defined on which do not contain surface subgroups. We prove that a presumably smaller class N' is closed under amalgamating along complete subgraphs, and also under adding bisimplicial edges. It follows that chordal graphs and chordal bipartite graphs belong to N'.

math.GR↗

arXiv2006

Co-contractions of Graphs and Right-angled Artin Groups

We define an operation on finite graphs, called co-contraction. By showing that co-contraction of a graph induces an injective map between right-angled Artin groups, we exhibit a family of graphs, without any induced cycle of length at least 5, such that the right-angled Artin groups on those graphs contain hyperbolic surface groups. This gives the negative answer to a question raised by Gordon, Long and Reid.

math.GR↗