Let $G=F\ast_\varphi t$ be an HNN extension of a free group $F$ with two equal associated normal subgroups $H_1 = H_2$ of finite index. We prove that the word problem in $G$ is decidable in polynomial time. This result extends to the case where the subgroups $H_1=H_2$ are not normal, provided that the isomorphism $\varphi:H_1\to H_2$ satisfies an additional condition described in Section 5.

翻译：设$G=F\ast_\varphi t$为自由群$F$的一个HNN扩张，其两个关联正规子群$H_1 = H_2$具有有限指数。我们证明$G$中的字问题可在多项式时间内判定。该结果可推广至子群$H_1=H_2$非正规的情形，前提是同构映射$\varphi:H_1\to H_2$满足第5节所述的附加条件。