# Full-featured peak reduction in right-angled Artin groups

@article{Day2014FullfeaturedPR,
title={Full-featured peak reduction in right-angled Artin groups},
author={Matthew B. Day},
journal={Algebraic \& Geometric Topology},
year={2014},
volume={14},
pages={1677-1743}
}
• M. Day
• Published 1 November 2012
• Mathematics
• Algebraic & Geometric Topology
We prove a new version of the classical peak reduction theorem for automorphisms of free groups in the setting of right-angled Artin groups. We use this peak reduction theorem to prove two important corollaries about the action of the automorphism group of a right-angled Artin group AA on the set of k ‐tuples of conjugacy classes from AA : orbit membership is decidable, and stabilizers are finitely presentable. Further, we explain procedures for checking orbit membership and building…
14 Citations

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## References

SHOWING 1-10 OF 26 REFERENCES
Peak reduction and finite presentations for automorphism groups of right-angled Artin groups
We generalize the peak reduction algorithm (Whitehead’s theorem) for free groups to a theorem about a general right-angled Artin group AΓ. As an application, we find a finite presentation for the
Outer space for right-angled Artin groups I
• Mathematics
• 2012
For a general right-angled Artin group A_G we introduce an Outer space O_G; this is a contractible, finite-dimensional space with a proper action of the outer automorphism group Out(A_G). In the
Symplectic structures on right-angled Artin groups: Between the mapping class group and the symplectic group
We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g, Z). We then prove that these groups are
An introduction to right-angled Artin groups
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
Some finitely presented subgroups of the automorphism group of a free group
The aim of this paper is to show that certain “natural” subgroups of the automorphism group ~2 of a finitely generated free group F have finite presentations. It follows from our results, as a
Graph groups are biautomatic
Abstract Graph groups admit a (finite) presentation in which each relation is of the form xy = yx for generators x and y . While the two extreme cases of graph groups, free groups and free abelian
Algorithms and Geometry for Graph Products of Groups
• Mathematics
• 1995
Recent work of Gromov, Epstein, Cannon, Thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups. (See [16],[17] and
On Systems of Equations over Free Partially Commutative Groups
• Mathematics
• 2008
Using an analogue of Makanin-Razborov diagrams, the authors give an effective description of the solution set of systems of equations over a partially commutative group (right-angled Artin group) G.
On Equivalent Sets of Elements in a Free Group
together with the 'simple automorphism' which replaces a, by its inverse. Relative to this kind of equivalence we have very little to add to a paper by J. Nielsen,2 in which he gives a mechanical
Automorphisms of graph groups
Abstract Given a graph Γ = ( V , E ), the graph group on Γ is the group generated by the vertex set V with the defining relations being that two adjacent vertices commute. In the following, the