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}'_{<}.
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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