4 problems

For $n\geq 2$, consider the balanced presentation … Its relators are $(x^ny^{-n-1},xyxy^{-1}x^{-1}y^{-1})$, and its generators are $(x,y)$. Akbulut–Kirby conjecture. The relator pa…

Finite-presentation isolated-groups conjecture. The set of isolated groups is not $\Lambda_{FP}$-semi-decidable.

Lower central series rank conjecture. The ranks satisfy

Let $P=\left\langle X\mid R\right\rangle$ be a finite presentation of a non-abelian group $\overline{P}$, and let $W(X)$ denote the words in the generators $X$. For words…