The statement “no nonabelian simple group can be obtained from a nonsimple group by adding one generator and one relator” 1) is equivalent to the Kervaire–Laudenbach conjecture; 2) becomes true under the additional assumption that the initial nonsimple group is either finite or torsion-free.

Coalgebras of set functors preserving weak pullbacks are particularly well behaved Invoking a result by Carboni Kelly and Wood we show that this can be explained by the fact that such functors can be uniquely extended to a relator This insight next suggests a de nition of metric bisimulation

Theorem 1 The word problem for any 1-relator monoids can be reduced to the left side divisibility problem for monoids M presented in 2 generators by 1 defining relation of the form aU = bV . For the solution of this problem it sufficies to find an algorithm to recognize for any word aW (or for any word bW ) whether or not it is left side divisible in M by the letter b (accordingly by the letter...

The principal result of this paper is a reduction of the word problem for one relator semigroups. Crudely, a relator (w,w′) on the alphabet X is achievable if the letters in the relator can actively participate in sequences of transitions for the semigroup presentation 〈X ; (w,w′)〉. (A precise definition of achievable relators is given in section 3.) Empirically, achievable relators are rare. W...