Thomas Koberda

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 A , we produce a new graph through a purely combinatorial procedure, and call it the extension graph A e of A . We produce a second graph A e k , the clique graph of A e , by adding an extra vertex for each complete subgraph of A e . We prove that each finite induced subgraph E of A e gives rise to an inclusion A.E/! A.A/. Conversely, we show that if there is an inclusion A.E/! A.A/ then E is an induced subgraph of A e k . These results have a number of corollaries. Let P4 denote the path on four vertices and let Cn denote the cycle of length n. We prove that A.P4/ embeds in A.A/ if and only if P4 is an induced subgraph of A . We prove that if F is any finite forest then A.F/ embeds in A.P4/. We recover the first author’s result on co-contraction of graphs, and prove that if A has no triangles and A.A/ contains a copy of A.Cn/ for some n 5, then A contains a copy of Cm for some 5 m n. We also recover Kambites’ Theorem, which asserts that if A.C4/ embeds in A.A/ then A contains an induced square. We show that whenever A is triangle-free and A.E/ < A.A/ then there is an undistorted copy of A.E/ in A.A/. Finally, we determine precisely when there is an inclusion A.Cm/!A.Cn/ and show that there is no “universal” two‐dimensional right-angled Artin group. 20F36

The geometry of the curve graph of a right-angled Artin group

We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we are able to develop a Nielsen--Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. On the geometric side, we establish an analogue of Masur and Minskys Bounded Geodesic Image Theorem and their distance formula.

Anti-trees and right-angled Artin subgroups of braid groups

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, and, consequently, into a pure braid group. It follows that G is a quasi-isometrically embedded subgroup of the area-preserving diffeomorphism groups of the 2‐disk and of the 2‐sphere with L p ‐metrics for suitable p . Another corollary is that there exists a closed hyperbolic manifold group of each dimension which admits a quasi-isometric group embedding into a pure braid group. Finally, we show that the isomorphism problem, conjugacy problem, and membership problem are unsolvable in the class of finitely presented subgroups of braid groups.

Right-angled Artin groups in the C ∞ diffeomorphism group of the real line

We prove that every right-angled Artin group embeds into the C∞ 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∞ diffeomorphism group of the real line.

Effective Algebraic Detection of the Nielsen–Thurston Classification of Mapping Classes

In this paper, we propose two algorithms for determining the Nielsen–Thurston classification of a mapping class ψ on a surface S. We start with a finite generating set X for the mapping class group and a word ψ in 〈X〉. We show that if ψ represents a reducible mapping class in Mod(S), then ψ admits a canonical reduction system whose total length is exponential in the word length of ψ. We use this fact to find the canonical reduction system of ψ. We also prove an effective conjugacy separability result for π1(S) which allows us to lift the action of ψ to a finite cover of S whose degree depends computably on the word length of ψ, and to use the homology action of ψ on to determine the Nielsen–Thurston classification of ψ.

Effective Indices of Subgroups in Baumslag–Pride Groups with Free Quotients

Given a group with at least two more generators than relations, we give an effective estimate on the minimal index of a subgroup with a nonabelian free quotient. We show that the index is bounded by a polynomial in the length of the relator words. We also provide a lower bound on the index.

Free products and the algebraic structure of diffeomorphism groups

Let

RIGHT-ANGLED ARTIN GROUPS AND A GENERALIZED ISOMORPHISM PROBLEM FOR FINITELY GENERATED SUBGROUPS OF MAPPING CLASS GROUPS

M

Asymptotic linearity of the mapping class group and a homological version of the Nielsen–Thurston classification

be a compact one--manifold, and let

The geometry of purely loxodromic subgroups of right-angled Artin groups

\mathrm{Diff}^{1+\mathrm{bv}}(M)