On parameterized complexity of the word search problem in the Baumslag-Gersten group.

We consider the word search problem in the Baumslag-Gersten group GB. We show that the parameterized complexity of this problem, where the area of van Kampen diagram serves as a parameter, is polynomial in the length of the input and the parameter. This contrasts the well-known result that the Dehn function and the time complexity of the word search problem in GB are non-elementary.