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We introduce a method of finding large non-positively curved subcomplexes in certain spherical Deligne complexes, which is effective for studying fillings of certain 6-cycles in spherical Deligne complexes. As applications, we show the…

We show that for each positive integer $k$ there exist right-angled Artin groups containing free-by-cyclic subgroups whose monodromy automorphisms grow as $n^k$. As a consequence we produce examples of right-angled Artin groups containing…

In this paper, we show that every irreducible $2$-dimensional Artin group $A_{\Gamma}$ of rank at least $3$ is acylindrically hyperbolic. We do this by studying the action of $A_{\Gamma}$ on its modified Deligne complex. Along the way, we…

We prove using a novel random matrix model that all right-angled Artin groups have a sequence of finite dimensional unitary representations that strongly converge to the regular representation. We deduce that this result applies also to:…

We show that the class of large-type Artin groups is invariant under isomorphism, in stark contrast with the corresponding situation for Coxeter groups. We obtain this result by providing a purely algebraic characterisation of large-type…

Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical complexes with a variety of applications.…

We prove that groups in a certain class of metabelian locally compact groups, have quadratic Dehn function. As an application, we embed the solvable Baumslag-Solitar groups into finitely presented metabelian groups with quadratic Dehn…

We prove the Farrell-Jones fibered isomorphism conjecture for several classes of Artin groups of finite and affine types. As a consequence, we compute explicitly the surgery obstruction groups of the finite type pure Artin groups.

We show that many $2$-dimensional Artin groups are residually finite. This includes $3$-generator Artin groups with labels $\geq 4$ except for $(2m+1, 4,4)$ for any $m\geq 2$. As a first step towards residual finiteness we show that these…

We consider a special class of framed links that arise from the hexatangle. Such links are introduced in [arXiv:0807.1677], which was also analyzed when the 3-manifold obtained after performing integral Dehn surgery on closed pure 3-braids…

We prove that Abels' group over an arbitrary nondiscrete locally compact field has a quadratic Dehn function. As applications, we exhibit connected Lie groups and polycyclic groups whose asymptotic cones have uncountable abelian fundamental…

We prove that the Bridson-Dison group has quartic Dehn function, thereby providing the first precise computation of the Dehn function of a subgroup of a direct product of free groups with super-quadratic Dehn function. We also prove that…

In this note we give the quasi-isometry classification for a class of right angled Artin groups. In particular, we obtain the first such classification for a class of Artin groups with dimension larger than 2; our families exist in every…

We study the geometry of nonrelatively hyperbolic groups. Generalizing a result of Schwartz, any quasi-isometric image of a non-relatively hyperbolic space in a relatively hyperbolic space is contained in a bounded neighborhood of a single…

We develop new methods for computing the precise Dehn functions of coabelian subgroups of direct products of groups, that is, subgroups which arise as kernels of homomorphisms from the direct product onto a free abelian group. These improve…

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure…

For a finitely generated group $G$ and collection of subgroups $\mathcal{P}$ we prove that the relative Dehn function of a pair $(G,\mathcal{P})$ is invariant under quasi-isometry of pairs. Along the way we show quasi-isometries of pairs…

The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the…

This is a survey of the recent work in algorithmic and asymptotic properties of groups. I discuss Dehn functions of groups, complexity of the word problem, Higman embeddings, and constructions of finitely presented groups with extreme…

Let $G$ be a right-angled Artin group with $|\mathrm{Out}(G)|<+\infty$. We prove that if a countable group $H$ with bounded torsion is measure equivalent to $G$, with an $L^1$-integrable measure equivalence cocycle towards $G$, then $H$ is…