Narrow Results

We determine the restricted Lie algebra associated with the Zassenhauss filtration of a right-angled Artin group. We also provide another proof for a result of Koberda saying that the graph is uniquely determined by the first and second cohomology groups, together with the cup product, of the associated right-angled Artin group.

We correct the stability bound in our classification of closed, spin${}^{+}$, topological four-manifolds with fundamental group $\pi$ of cohomological dimension $\le 3$ (up to $s$-cobordism), after stabilization by connected sum with copies of $S^2\times S^2$. If $\pi$ is a right-angled Artin group whose defining graphs have no four-cliques, then the new stability bound is $r\ge max(b_3(\pi ),6)$. The other results of the paper are not affected.

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.

In this paper, we generalize Magnus' Freiheitssatz and solution to the word problem for one-relator groups by considering one-relator quotients of certain classes of right-angled Artin groups and graph products of locally indicable polycyclic groups.

We show that a right-angled Artin group, defined by a graph Γ that has at least three vertices, does not split over an infinite cyclic subgroup if and only if Γ is biconnected. Further, we compute JSJ-decompositions of 1-ended right-angled Artin groups over infinite cyclic subgroups.

For a finite simplicial graph $\mathrm{\Gamma }$, let $A(\mathrm{\Gamma })$ denote the right-angled Artin group on $\mathrm{\Gamma }$. Recently, Kim and Koberda introduced the extension graph ${\mathrm{\Gamma }}^e$ for $\mathrm{\Gamma }$, and established the Extension Graph Theorem: for finite simplicial graphs ${\mathrm{\Gamma }}_1$ and ${\mathrm{\Gamma }}_2$, if ${\mathrm{\Gamma }}_1$ embeds into ${\mathrm{\Gamma }}_2^e$ as an induced subgraph then $A({\mathrm{\Gamma }}_1)$ embeds into $A({\mathrm{\Gamma }}_2)$. In this paper, we show that the converse of this theorem does not hold for the case ${\mathrm{\Gamma }}_1$ is the complement of a tree and for the case ${\mathrm{\Gamma }}_2$ is the complement of a path graph.

The automorphism group of a partially commutative group ${𝔾}_{\mathrm{\Gamma }}$ is generated by four types of automorphism: graph automorphisms, inversions, transvections and vertex conjugating automorphisms. We find a characterisation of the subgroup $\mathrm{\text{S}}t({𝒦})$ generated by inversions and transvections, in terms of stabilisers of subgroups of ${𝔾}_{\mathrm{\Gamma }}$ generated by the so-called “admissible” subsets of $V(\mathrm{\Gamma })$. This is used to give a decomposition of $\mathrm{\text{S}}t({𝒦})$ as a chain of semi-direct products of (mostly) tractable and recognisable subgroups; which in turn gives rise to a presentation of $\mathrm{\text{S}}t({𝒦})$.

We show that if a right-angled Artin group $A(\mathrm{\Gamma })$ has a non-trivial, minimal action on a tree $T$ with more than two ends, then $\mathrm{\Gamma }$ contains a separating subgraph $\mathrm{\Lambda }$ such that $A(\mathrm{\Lambda })$ stabilizes an edge in $T$.

The action of a right-angled Artin group on its extension graph is known to be acylindrical because the cardinality of the so-called r-quasi-stabilizer of a pair of distant points is bounded above by a function of r. The known upper bound of the cardinality is an exponential function of r. In this paper we show that the r-quasi-stabilizer is a subset of a cyclic group and its cardinality is bounded above by a linear function of r. This is done by exploring lattice theoretic properties of group elements, studying prefixes of powers and extending the uniqueness of quasi-roots from word length to star length. We also improve the known lower bound for the minimal asymptotic translation length of a right-angled Artin group on its extension graph.

Let $UV{B}_n$ and $UV{P}_n$ be the unrestricted virtual braid group and the unrestricted virtual pure braid group on n strands, respectively. We study the groups $UV{B}_n$ and $UV{P}_n$, and our main results are as follows: for $n\ge 5$, we give a complete description, up to conjugation, to all possible homomorphisms from $UV{B}_n$ to the symmetric group $S_n$. For $n\ge 3$, we characterize all possible images of $UV{B}_n$, under a group homomorphism, to any finite group $G$. For $n\ge 5$, we prove that $UV{P}_n$ is a characteristic subgroup of $UV{B}_n$. In addition, we determine the automorphism group of $UV{P}_n$ and we prove that ${\mathbb{Z}}_2\times {\mathbb{Z}}_2$ is a subgroup of the outer automorphism group of $UV{B}_n$. Lastly, we show that $UV{B}_n$ and $UV{P}_n$ are residually finite and Hopfian but not co-Hopfian. We also remark that some of these results hold accordingly for the welded braid group $W{B}_n$ and we discuss about its automorphism group.

In this paper, we show that every right-angled Artin group embeds into the $n$-dimensional Thompson group $nV$ for some $n$, by constructing a set of embeddings. This is an improvement of the previous result of Belk, Bleak and Matucci, which gives a different set of embeddings from right-angled Artin groups into higher-dimensional Thompson groups. Compared to their construction, the dimension $n$ of the target group is always smaller with our construction.

We classify closed, spin${}^{+}$, topological $4$-manifolds with fundamental group $\pi$ of cohomological dimension $\le 3$ (up to $s$-cobordism), after stabilization by connected sum with at most $b_3(\pi )$ copies of $S^2\times S^2$. In general, we must also assume that $\pi$ satisfies certain $K$-theory and assembly map conditions. Examples for which these conditions hold include the torsion-free fundamental groups of $3$-manifolds and all right-angled Artin groupswhose defining graphs have no $4$-cliques.

The purpose of this paper is to give a characterization of families of expander graphs via right-angled Artin groups. We prove that a sequence of simplicial graphs ${\{{\mathrm{\Gamma }}_i\}}_{i\in \mathbb{N}}$ forms a family of expander graphs if and only if a certain natural mini-max invariant arising from the cup product in the cohomology rings of the groups ${\{A({\mathrm{\Gamma }}_i)\}}_{i\in \mathbb{N}}$ agrees with the Cheeger constant of the sequence of graphs, thus allowing us to characterize expander graphs via cohomology. This result is proved in the more general framework of vector space expanders, a novel structure consisting of sequences of vector spaces equipped with vector-space-valued bilinear pairings which satisfy a certain mini-max condition. These objects can be considered to be analogues of expander graphs in the realm of linear algebra, with a dictionary being given by the cup product in cohomology, and in this context represent a different approach to expanders that those developed by Lubotzky–Zelmanov and Bourgain–Yehudayoff.