On the conjugacy problem in group $\bf F/{N_1\cap N_2}$

TL;DR

This paper proves that the conjugacy problem is solvable in certain free group quotients when the combined relations satisfy specific small cancellation conditions and the presentation is atorical, extending known results.

Contribution

It establishes solvability of the conjugacy problem in free group quotients under combined small cancellation and atoricality conditions, generalizing previous results.

Findings

Conjugacy problem solvable under C(6) with atorical presentation.

Conjugacy problem solvable under C(7) conditions.

Extension of solvability results to intersections of normal closures.

Abstract

Let $N_1$ (resp., $N_2$) be the normal closure of a finite symmetrized set $R_1$ (resp., $R_2$) of a finitely generated free group $F = F(A)$. It is well-known that if $R_i$ satisfies the condition C(6), then the conjugacy problem is solvable in $F/N_i$. In the present paper we prove that if $R_1\cup R_2$ satisfies the condition C(6) and the presentation $<A\, \mid R_1,R_2>$ is atorical, then the conjugacy problem is solvable in $F/{N_1\cap N_2}$. In particular, if $R_1\cup R_2$ satisfies the condition C(7) then the conjugacy problem is solvable in $F/{N_1\cap N_2}$.