arXiv research

The study examines subgroups of RACGs and RAAGs, focusing on their RAAG properties.

Characterizes groups arising as fixed subgroups of RAAG automorphisms.

Graphs with specific spanning trees yield RAAGs, with applications to BBGs.

Study on representation varieties of RAAGs for different Lie groups.

We prove that every RAAG (a Right-Angled Artin Group) embeds in the group of Hamiltonian symplectomorphisms of the 2-sphere.

Proves a representation that separates a subgroup in RAAGs.

Study on stable commutator length in RAAGs and Coxeter groups, proving spectral gaps and hardness results.

In this paper we prove that RAAGs are distinguished from each other by their pro-$p$ completions for any choice of prime $p$, and that RACGs are distinguished from each other by their pro-2 completions. We also give a new proof that hyperbolic virtually special groups are good in the sense of Serre. Furthermore we give…

We are motivated by the question that for which class of right-angled Artin groups (RAAG's), the quasi-isometry classification coincides with commensurability classification. This is previously known for RAAG's with finite outer automorphism groups. In this paper, we identify two classes of RAAG's, where their outer au…

Extended characterization of RAAGs with zero minimal volume entropy.

The paper studies automorphisms of RAAGs and RACGs, proving properties of their fixed subgroups.

Study shows shorter words for group elements in surface groups and RAAGs.

Finite groups of RAAGs act on a contractible space without fixed points.

New proof shows lower bound for commutator length in RAAGs.

Outer space for RAAGs is a contractible finite-dimensional space for automorphisms.

We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperb…

Investigates minimal genus of second homology classes in RAAGs, finding bounds and specific cases.

Study shows RAAG automorphisms and outer automorphisms are not relatively hyperbolic.

This paper focuses on tools for constructing 4-manifolds that have fundamental group $G$ isomorphic to a right-angled Artin group and that are also minimal, in the sense that they minimize $b_2(M)$, the dimension of $H_2(M;\mathbb{Q})$. For a finitely presented group $G$, define $h(G) = \min\{ b_2(M) | M \in \mathcal M…

We study the outer automorphism group of a right-angled Artin group $A_Γ$ with finite defining graph $Γ$. We construct a subnormal series for $Out(A_Γ)$ such that each consecutive quotient is either finite, free-abelian, $GL(n,\mathbb{Z})$, or a Fouxe-Rabinovitch group. The last two types act respectively on a symmetri…

New contractible complex shows virtual cohomological dimension of RAAGs.

Study on symmetric automorphisms of RAAGs, proving finiteness properties and contractibility.

We show that the homology of the automorphism group of a right-angled Artin group stabilizes under taking products with any right-angled Artin group.

Motivated by the notion of cusp excursion in geometrically finite hyperbolic manifolds, we define a notion of excursion in any subgroup of a given group, and study its asymptotic distribution for right-angled Artin groups and graph products. In particular, for any irreducible right-angled Artin group we show that with …

Study presentations of groups that can be generalised over continuous open group monomorphisms.

We characterize groups quasi-isometric to a right-angled Artin group $G$ with finite outer automorphism group. In particular all such groups admit a geometric action on a $CAT(0)$ cube complex that has an equivariant "fibering" over the Davis building of $G$.

For every finite graph $Γ$, we define a simplicial complex associated to the outer automorphism group of the RAAG $A_Γ$. These complexes are defined as coset complexes of parabolic subgroups of $Out^0(A_Γ)$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotop…

We prove that every right-angled Artin group embeds into the $C^{\infty}$ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the $C^{\infty}$ diffeomorphism group of the real line.

We prove Nielsen realisation for finite subgroups of the groups of untwisted outer automorphisms of RAAGs in the following sense: given any graph $Γ$, and any finite group $G\leqslant \mathrm{U}^0(A_Γ) \leqslant \mathrm{Out}^0(A_Γ)$, we find a non-positively curved cube complex with fundamental group $A_Γ$ on which $G$…

The paper studies graph products of groups and recovers graph and vertex groups under certain conditions.

In this article, we determine the function $\ell(S_{g, p})$ such that the right-angled Artin group $G(P_{m})$ is embedded in the mapping class group $\mathrm{Mod}(S_{g, p})$ if and only if $m$ is not more than $\ell(S_{g, p})$. Using this function and Birman--Hilden theory, we prove that $\mathrm{Mod}(S_{0, p})$ is vir…

New groups derived from square configurations have right-angled and HNN structures.

The study examines discrete curvature notions on Cayley graphs of certain groups.

Flawed groups are shown to include all finitely generated groups isomorphic to free products of nilpotent groups.

For a finite simplicial graph $Γ$, let $G(Γ)$ denote the right-angled Artin group on the complement graph of $Γ$. In this article, we introduce the notions of "induced path lifting property" and "semi-induced path lifting property" for immersions between graphs, and obtain graph theoretical criteria for the embedabilit…

Let $G$ be a right-angled Artin group with defining graph $Γ$ and let $H$ be a finitely generated group quasi-isometric to $G(Γ)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $Γ$ does not have induced 4-cycle; (3) $Γ$ is star-rigid; then $H$ is commensurable to $G$. We show condition (2) is…

We develop a new criterion to tell if a group $G$ has the maximal gap of $1/2$ in stable commutator length (scl). For amalgamated free products $G = A \star_C B$ we show that every element $g$ in the commutator subgroup of $G$ which does not conjugate into $A$ or $B$ satisfies $scl(g) \geq 1/2$, provided that $C$ embed…

We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" grap…

Unified algebraic framework for virtual braid structures with strong structural consequences.

We say a graph has property $\mathcal{P}_{g,p}$ when it is an induced subgraph of the curve graph of a surface of genus $g$ with $p$ punctures. Two well-known graph invariants, the chromatic and clique numbers, can provide obstructions to $\mathcal{P}_{g,p}$. We introduce a new invariant of a graph, the 'nested complex…