Condition \(\mathcal{T}'\) was introduced in [BMSC20]. We give here the definition in terms of van Kampen diagrams. If \(v\) is an interior vertex in a reduced diagram \(D\), let
\[d'_F(v)=\sum_{c\in v} \frac{\ell_1(c)+\ell_2(c)}{\ell_r(c)}\]
where the sum is over all corners at \(v\), \(\ell_r(c)\) is the length of the relator corresponding to the corner \(c\) and \(\ell_i(c)\) are the lenghts of the words written in the edges of the corner. A presentation \(P\) satisfies \(\mathcal{T}'\) if \(2\leq d(v)-d'_F(v)\) for every interior vertex \(v\) in every reduced diagram \(D\) over \(P\). If this inequality is strict we say that the presentation satisfies condition \(\mathcal{T}'_{<}\).
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If \(P\) is a finite presentations satisfying \(\mathcal{T}'_{<}-C(3)\) then the group presented by \(P\) is hyperbolic [BMSC20, Theorem 3.3]. A presentation without proper powers which satisfies condition \(\mathcal{T}'\) is diagrammatically reducible (DR) [BMSC20, Theorem 3.2]. If \(P\) satisfies \(\mathcal{T'}-C'(\frac{1}{2})\) and the defining relators have the same length then \(P\) satisfies a quadratic isoperimetric inequality and if in addition \(P\) is finite, it has solvable conjugacy problem [BMSC20, Theorem 4.2].
‣ GroupSatisfiesTauPrime( G ) | ( function ) |
Given an FpGroup G gives the minimum of \(d(v)-d'_F(v)\). Thus if the function returns a number greater than or equal to \(2\) the group presentation satisfies \(\mathcal{T}'\) and if the number is greater than \(2\) it satisfies \(\mathcal{T}'_{<}\). This function implements the algorithm in [BMSC20]. For the moment the computation is done numerically using floating point numbers (this may be modified in a future version to do the computation exactly with rational numbers). External binaries need to be compiled.
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