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The symmetrisation of a relator set closes under inverses and cyclic conjugates
Definition
Let be a presentation in which every relator in is cyclically reduced (Cyclically reduced words, Group presentation by generators and relations). The symmetrisation of is the set consisting of all cyclic conjugates of every relator and of every inverse word .
Thus a reduced word lies in exactly when there is an such that is a cyclic conjugate of or of . By construction is closed under taking inverses and under cyclic conjugation.
Depends on
Used by
- A piece is a common initial segment occurring in two distinct places of a symmetrised relator set Definition
- Dehn-reduced words and Dehn presentations Definition
- The condition T(q) forbids short cycles of pieces in the relator graph Definition
- Symmetrising a one-relator presentation adds the cyclic conjugates and inverse cyclic conjugates Example
- A relator set and its symmetrisation have the same normal closure Lemma
Dependency tree · two levels
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Sources
- GAP SmallCancellation manual, Chapter 1: Small Cancellation Theory — the classical conditions (standard reference, not scraped)
- Jay Williams, Universal Countable Borel Quasi-Orders (standard reference, not scraped)
- Nicholas Touikan, An Introduction to Combinatorial and Geometric Group Theory, Section 3.5 (standard reference, not scraped)
- Clara Löh, Geometric Group Theory: An Introduction, Section 7.4.1 (standard reference, not scraped)