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1 Small Cancellation Theory: the classical conditions

1.1 Introduction

The standard reference for Small Cancellation Theory is Lyndon-Schupp [LS01, Chapter V]. We review here some definitions and results.

A subset \(R\) of a free group \(F\) is called symmetrized if all elements of \(R\) are cyclically reduced and for each \(r\) in \(R\) all cyclically reduced conjugates of \(r\) and \(r^{-1}\) are also in \(R\). If the set \(R\) is not symmetrized we may work instead with the symmetrization \(R^*\) of \(R\) (the smallest symmetrized set containing \(R\)).

A piece of \(R\) is a word that is a common prefix of two different words in the symmetrized set \(R^*\).

We say that a presentation \(P=\langle X \mid R\rangle\) satisfies the small cancellation condition \(C(p)\) if no relator \( r \) in \(R^*\) is a product of fewer than \(p\) pieces.

We say that \(P\) satisfies the small cancellation condition \(C'(\lambda)\) if for all \( r \) in \(R^*\), if \(r=bc\) and \(b\) is a piece then \(|b| < \lambda |r| \).

We say that \(P\) satisfies the small cancellation condition \(T(q)\) if for all \(h\) such that \(3\leq h < q\) and for all elements \(r_1,\ldots, r_h\) in \(R^*\), if no succesive elements \(r_i,r_{i+1}\) is an inverse pair, then at least one of the products \(r_1r_2,\ldots r_{h-1}r_h, r_hr_1\) is reduced without cancellation. Condition \(T(q)\) may be rephrased in terms of the Whitehead graph of the presentation \(P\).

If a group \(G=\langle X \mid R\rangle\) satisfies \(C'(1/6)\) then Dehn's algorithm (see [LS01, Chapter V, Section 4]) solves the word problem for \(G\).

If a group \(G=\langle X \mid R\rangle\) satisfies \(C(6)\) or \(C(4)–T(4)\) or \(C(3)–T(6)\) then \(G\) has solvable word problem and solvable conjugacy problem (see [LS01, Chapter V, Sections 5 and 6]).

If a presentation \(P=\langle X \mid R\rangle\) satisfies \(C(6)\) and no relator of \(P\) is a proper power then the presentation complex of \(P\) is aspherical.

1.2 Pieces

1.2-1 PiecesOfGroup

Returns the set of pieces of the FpGroup G.

gap> F:=FreeGroup(["a","b"]);;
gap> AssignGeneratorVariables(F);;
#I  Assigned the global variables [ a, b ]
gap> r:=a*b*a*b^2*a*b^3;;
gap> G:=F/[r];;
gap> PiecesOfGroup(G);
[ a^-1, a, b^-1, b, a^-1*b^-1, a*b, b^-1*a^-1, b^-2, b*a, b^2, a^-1*b^-2, 
  a*b^2, b^-1*a^-1*b^-1, b^-2*a^-1, b*a*b, b^2*a, b^-1*a^-1*b^-2, 
  b^-2*a^-1*b^-1, b*a*b^2, b^2*a*b ]

1.2-2 PiecesOfPresentation

Returns the set of pieces of the presentation P.

1.3 Metric small cancellation conditions

1.3-1 GroupSatisfiesCPrime

Returns true if the FpGroup G satisfies the small cancellation condition C'(lambda), false otherwise. If the optional argument explain is true instead of returning false returns an explanation of this result.

gap> F:=FreeGroup(["a1","b1","a2","b2","a3","b3"]);;
gap> AssignGeneratorVariables(F);;
#I  Assigned the global variables [ a1, b1, a2, b2, a3, b3 ]
gap> r:=Comm(a1,b1)*Comm(a2,b2)*Comm(a3,b3);;
gap> G:=F/[r];;
gap> GroupSatisfiesCPrime(G,1/11);
true
gap> GroupSatisfiesCPrime(G,1/12);
false
gap> GroupSatisfiesCPrime(G,1/12,true);
[ false, a1^-1*b1^-1*a1*b1*a2^-1*b2^-1*a2*b2*a3^-1*b3^-1*a3*b3, 
  " starts with a piece of length ", 1 ]

1.3-2 PresentationSatisfiesCPrime

Returns true if the presentation P satisfies the small cancellation condition C'(lambda), false otherwise. If the optional argument explain is true instead of returning false returns an explanation of this result.

1.4 Non-metric small cancellation conditions

1.4-1 GroupSatisfiesC

Returns true if the FpGroup G satisfies the small cancellation condition C(p), false otherwise. If the optional argument explain is true instead of returning false returns an explanation of this result.

gap> GroupSatisfiesC(G,12);
true
gap> GroupSatisfiesC(G,13);
false
gap> GroupSatisfiesC(G,13,true);
[ false, a1^-1*b1^-1*a1*b1*a2^-1*b2^-1*a2*b2*a3^-1*b3^-1*a3*b3, 
  " is the product of pieces of the following lengths: ", 
  [ 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1 ] ]

1.4-2 PresentationSatisfiesC

Returns true if the presentation P satisfies the small cancellation condition C(p), false otherwise. If the optional argument explain is true instead of returning false returns an explanation of this result.

1.4-3 GroupSatisfiesT

Returns true if the FpGroup G satisfies the small cancellation condition T(q), false otherwise.

gap> GroupSatisfiesT(G,12);
true
gap> GroupSatisfiesT(G,13);
false

1.4-4 PresentationSatisfiesT

Returns true if the presentation P satisfies the small cancellation condition T(q), false otherwise.

1.5 Implementation details

Relators and their prefixes are stored in a data structure called trie [Wik19]. This provides an efficient way to write a word as a product of pieces using the minimum possible number of pieces.

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