Abstract. Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the onerelator Baumslag group is in P. Before that the best known upper bound has been non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic algebra and algorithmic group theory: 1. We define a modified reduction procedure on power circuits which runs in quadratic time thereby improving the known cubic time complexity. The improvement is crucial for our other results. 2. We improve the complexity of the Word Problem for the Baumslag group to cubic time thereby providing the first practical algorithm for that problem. 3. The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of power circuits.

We introduce the peak normal form for elements of the Baumslag-Solitar groups BS(p, q). This normal form is very close to the lengthlexicographical normal form, but more symmetric. Both normal forms are geodesic. This means the normal form of an element u −1 v yields the shortest path between u and v in the Cayley graph. For horocyclic elements the peak normal form and the length-lexicographical normal form coincide. The main result of this paper is that we can compute the peak normal form in polynomial time if p divides q. As consequence we can compute geodesic lengths in this case. In particular, this gives a partial answer to Question 1 in [4].For arbitrary p and q it is possible to compute the peak normal form (length-lexicographical normal form resp.) also the for elements in the horocyclic subgroup and, more generally, for elements which we call hills. This approach leads to a linear time reduction of the problem of computing geodesics to the problem of computing geodesics for Britton-reduced words where the t-sequence starts with t −1 and ends with t. *

This paper continues the 2012 STACS contribution by Diekert, Ushakov, and the author. We extend the results published in the proceedings in two ways.First, we show that the data structure of power circuits can be generalized to work with arbitrary bases q ≥ 2. This results in a data structure that can hold huge integers, arising by iteratively forming powers of q. We show that the properties of power circuits known for q = 2 translate to the general case. This generalization is non-trivial and additional techniques are required to preserve the time bounds of arithmetic operations that were shown for the case q = 2.The extended power circuit model permits us to conduct operations in the Baumslag-Solitar group BS(1, q) as efficiently as in BS(1, 2). This allows us to solve the word problem in the generalization H4(1, q) of Higman's group, which is an amalgamated product of four copies of the Baumslag-Solitar group BS(1, q) rather than BS(1, 2) in the original form.As a second result, we allow arbitrary numbers f ≥ 4 of copies of BS(1, q), leading to an even more generalized notion of Higman groups H f (1, q). We prove that the word problem of the latter can still be solved within the O(n 6 ) time bound that was shown for H4(1, 2).

Power circuits are data structures which support efficient algorithms for highly compressed integers. Using this new data structure it has been shown recently by Myasnikov, Ushakov and Won that the Word Problem of the onerelator Baumslag group is in P. Before that the best known upper bound has been non-elementary. In the present paper we provide new results for power circuits and we give new applications in algorithmic algebra and algorithmic group theory: 1. We define a modified reduction procedure on power circuits which runs in quadratic time thereby improving the known cubic time complexity. The improvement is crucial for our other results. 2. We improve the complexity of the Word Problem for the Baumslag group to cubic time thereby providing the first practical algorithm for that problem. 3. The main result is that the Word Problem of Higman's group is decidable in polynomial time. The situation for Higman's group is more complicated than for the Baumslag group and forced us to advance the theory of power circuits.

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