On a conjecture in Artin groups

2021 ◽  
pp. 1-25
Author(s):  
Arye Juhász
Keyword(s):  
Artin Group ◽  
Two Dimensional ◽  
Artin Groups ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Infinite Type ◽  
Trivial Center

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

THE TITS CONJECTURE FOR LOCALLY REDUCIBLE ARTIN GROUPS

2000 ◽  
Vol 10 (06) ◽  
pp. 783-797 ◽  
Author(s):  
RUTH CHARNEY
Keyword(s):  
Artin Group ◽  
Artin Groups ◽  
Infinite Type ◽  
System A ◽  
Special Subgroup

Given an Artin system (A,S), a conjecture of Tits states that the subgroup A(2) of A generated by the squares of the generators in S is subject only to the obvious commutator relations between generators. In particular, A(2) is a right-angled Artin group. We prove this conjecture for a class of infinite type Artin groups, called locally reducible Artin groups, for which the associated Deligne complex has a CAT(0) geometry. We also prove that for any special subgroup AT of A, A(2)∩AT=(AT)(2).

scholarly journals Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

2021 ◽  
pp. 1-22
Author(s):  
Martín Axel Blufstein ◽  
Elías Gabriel Minian ◽  
Iván Sadofschi Costa
Keyword(s):  
Free Group ◽  
Positive Curvature ◽  
Satisfying Condition ◽  
Conjugacy Problem ◽  
Artin Group ◽  
Two Dimensional ◽  
Small Cancellation ◽  
Dehn Functions ◽  
Trivial Element ◽  
Standard Presentation

We present a metric condition $\TTMetric$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition $\TTMetric$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular, we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, $\TTMetric$ -groups have quadratic Dehn functions and solvable conjugacy problem. In the spirit of Greendlinger's lemma, we prove that if a presentation P = 〈X| R〉 of group G satisfies conditions $\TTMetric -C'(\frac {1}{2})$ , the length of any nontrivial word in the free group generated by X representing the trivial element in G is at least that of the shortest relator. We also introduce a strict metric condition $\TTMetricStrict$ , which implies hyperbolicity.

scholarly journals Graphical splittings of Artin kernels

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Enrique Miguel Barquinero ◽  
Lorenzo Ruffoni ◽  
Kaidi Ye
Keyword(s):  
Free Group ◽  
Artin Group ◽  
Artin Groups ◽  
Graphs Of Groups ◽  
Underlying Graph ◽  
Block Graphs ◽  
Right Angled Artin Groups

Abstract We study Artin kernels, i.e. kernels of discrete characters of right-angled Artin groups, and we show that they decompose as graphs of groups in a way that can be explicitly computed from the underlying graph. When the underlying graph is chordal, we show that every such subgroup either surjects to an infinitely generated free group or is a generalized Baumslag–Solitar group of variable rank. In particular, for block graphs (e.g. trees), we obtain an explicit rank formula and discuss some features of the space of fibrations of the associated right-angled Artin group.

When does a right-angled Artin group split over ℤ?

2014 ◽  
Vol 24 (06) ◽  
pp. 815-825 ◽  
Author(s):  
Matt Clay
Keyword(s):  
Cyclic Subgroup ◽  
Artin Group ◽  
Artin Groups ◽  
Cyclic Subgroups ◽  
Right Angled Artin Groups ◽  
Jsj Decompositions

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.

scholarly journals Embeddability of right-angled Artin groups on complements of trees

2018 ◽  
Vol 28 (03) ◽  
pp. 381-394
Author(s):  
Eon-Kyung Lee ◽  
Sang-Jin Lee
Keyword(s):  
Artin Group ◽  
Artin Groups ◽  
Graph Theorem ◽  
Induced Subgraph ◽  
Path Graph ◽  
Right Angled Artin Groups ◽  
The Right

For a finite simplicial graph [Formula: see text], let [Formula: see text] denote the right-angled Artin group on [Formula: see text]. Recently, Kim and Koberda introduced the extension graph [Formula: see text] for [Formula: see text], and established the Extension Graph Theorem: for finite simplicial graphs [Formula: see text] and [Formula: see text], if [Formula: see text] embeds into [Formula: see text] as an induced subgraph then [Formula: see text] embeds into [Formula: see text]. In this paper, we show that the converse of this theorem does not hold for the case [Formula: see text] is the complement of a tree and for the case [Formula: see text] is the complement of a path graph.

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