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AN ALGORITHM DETECTING DEHN PRESENTATIONS. G. N. ARZHANTSEVA
 

Summary: AN ALGORITHM DETECTING DEHN PRESENTATIONS.
G. N. ARZHANTSEVA
Abstract. An algorithm is given detecting whether or not a finite presenta­
tion of a group is a Dehn presentation (i.e. admitting Dehn's algorithm for the
word problem) with a certain condition.
Because being word hyperbolic is a Markov property of groups there cannot
exist an effective procedure for determining if a finitely presented group admits a
Dehn presentation (see, for example, [9]). However, there may exist an algorithm
to decide whether a finite presentation of a group is a Dehn presentation. In this
article we prove a result in this direction.
Definition 1. Let 1
2 Ÿ ff ! 1. We call a presentation hX j Ri of a group G
an ff­Dehn presentation if any non­empty freely reduced word w representing the
identity in G contains, as a subword, a word u which is also a subword of a cyclic
shift of some r 2 R \Sigma1 with juj ? ffjrj.
A Dehn presentation in the traditional sense is an ff­Dehn presentation with
ff = 1=2. Observe also that any ff­Dehn presentation is a Dehn presentation.
A Dehn presentation of a group G leads to a known Dehn's algorithm solving
the word problem for G. We refer to [5] for an interesting discussion and problems
on other Dehn's algorithms and different notions of Dehn presentations.

  

Source: Arzhantseva, Goulnara N. - Section de Mathématiques, Université de Genève

 

Collections: Mathematics