Small cancellation theory with non-homogeneous geometrical conditions and application to certain Artin groups

2012 ◽  
pp. 235-241
Author(s):  
A. Juhasz
Keyword(s):  
Artin Groups ◽  
Small Cancellation ◽  
Small Cancellation Theory

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.

On small cancellation theory over H.N.N. extensions

1983 ◽  
Vol 94 (1-2) ◽  
pp. 25-47 ◽  
Author(s):  
J. Perraud
Keyword(s):  
Word Problem ◽  
Finite Subset ◽  
Decision Problems ◽  
Normal Closure ◽  
Small Cancellation ◽  
Free Products ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Amalgamated Free Products ◽  
Small Cancellation Condition

SynopsisSmall cancellation theory has been extended to symmetrized subsets of free products, amalgamated free products and Higman-Neumann-Neumann (H.N.N.) extensions. We though that it was possible to obtain results on decision problems if we could define small cancellation conditions for finite subsets.Sacerdote and Schupp (1974) defined the small cancellation condition C'(l/6) for symmetrized subsets of an H.N.N. extension. We define this condition for finite subsets, with the following properties:For each finite subset X, there is a symmetrized subset X1 with the same normal closure and, if X1 satisfies C'(l/6), then X satisfies C'(l/6).For some H.N.N. extensions, we can decide whether any finite subset satisfies C'(l/6), and, in this case, we can solve the word problem for the corresponding quotient.

scholarly journals On the geometry of burnside quotients of torsion free hyperbolic groups

2014 ◽  
Vol 24 (03) ◽  
pp. 251-345 ◽  
Author(s):  
Rémi Coulon
Keyword(s):  
Hyperbolic Groups ◽  
Geometrical Approach ◽  
Small Cancellation ◽  
Small Cancellation Theory ◽  
Burnside Groups ◽  
Torsion Free

In this paper, we detail the geometrical approach of small cancellation theory used by Delzant and Gromov to provide a new proof of the infiniteness of free Burnside groups and periodic quotients of torsion-free hyperbolic groups.

scholarly journals GRAPH SMALL CANCELLATION THEORY APPLIED TO ALTERNATING LINK GROUPS

2012 ◽  
Vol 21 (11) ◽  
pp. 1250113
Author(s):  
RÉMI CUNÉO ◽  
HAMISH SHORT
Keyword(s):  
Small Cancellation ◽  
Cancellation Condition ◽  
Small Cancellation Theory ◽  
Link Group ◽  
Finite Sums ◽  
Small Cancellation Condition ◽  
Alternating Links ◽  
Alternating Link

We show that the Wirtinger presentation of a prime alternating link group satisfies a generalized small cancellation condition. This new version of Weinbaum's solution to the word and conjugacy problems for these groups easily extends to finite sums of alternating links.

FANS AND LADDERS IN SMALL CANCELLATION THEORY

2002 ◽  
Vol 84 (3) ◽  
pp. 599-644 ◽  
Author(s):  
JONATHAN P. McCAMMOND ◽  
DANIEL T. WISE
Keyword(s):  
Subject Classification ◽  
Small Cancellation ◽  
Classification Result ◽  
Small Cancellation Theory

This paper provides a strengthening of the theorems of small cancellation theory. It is proven that disc diagrams contain 'fans' of consecutive 2-cells along their boundaries. The size of these fans is linked to the strength of the small cancellation conditions satisfied by the diagram. A classification result is proven for disc diagrams satisfying small cancellation conditions. Any disc diagram either contains three fans along its boundary, or it is a ladder, or it is a wheel. Similar statements are proven for annular diagrams. 2000 Mathematical Subject Classification: 20F06.

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