... Reinert¹

The author was supported by the Deutsche Forschungsgemeinschaft (DFG).

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... convergent²

Convergent presentations for groups are string rewriting systems which are terminating and confluent.

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... ideals³

An ideal is called binomial if it has a basis solely consisting of polynomials of the form m₁ -m₂ where m₁,m₂ are monomials.

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...${\mathcal{F}}$⁴

Notice that while $\lambda$ is minimal with respect to $\leq$, the ordering is not compatible with multiplication as $\lambda < w$ then would imply $\lambda \circ w^{-1} = w^{-1} < w \circ w^{-1} = \lambda$.

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...v⁵

Those familiar with string rewriting systems should notice that prefix reducing a word u with a binomial $\ell-r$ where $\ell>r$ directly corresponds to prefix string reducing u with a rule $(\ell,r)$ followed by free reduction.

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... algorithm⁶

The FGLM Algorithm has been generalized to the setting of finitely presented groups in [5].

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... reduction⁷

We say a free reduced word $w \in {\mathcal{F}}$ prefix reduces to v (modulo free group reduction) using a rule $\ell \longrightarrow r$ if there exists $x \in {\mathcal{F}}$ such that $w \equiv\ell x$ and $v = r \circ x$.

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... rule⁸

Notice that there are trivial rules among these where the left and right hand sides coincide as words and these of course have to be removed in order to make the system terminating.

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... computed⁹

The steps in the computation of the prefix Gröbner basis can be directly related to Nielsen transformations (see [23]).

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... basis¹⁰

The computation of the prefix Gröbner basis is related to the filling of the first line of the tables in TC and the deduction of equations

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...$H = \{ \tau \ast(r -1) \mid r \in R \}$¹¹

The set H realizes the addition of subgroup generators $\tau \circ r \circ\tau^{-1}$ or in TC corresponds to marking the first and last slot of each relator table with the newly found coset representative $\tau$.

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... subgroup¹²

This corresponds to the coset collapses in TC.

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