Computation in word-hyperbolic groups (invited talk abstract)

There is an important class of finitely presented groups known as <italic>automatic groups</italic>, in which many computations can be undertaken efficiently. In particular, there is a normal form for group elements which is the language of a finite state automation, and arbitrary words in the group generators can be reduced to normal form in quadratic time. The normal form can also be used to enumerate group elements, and to compute the (rational) growth function of the group. The theory of automatic groups is developed [2], and the algorithms are described in more detail in [4]. Implementations of the algorithms are available in a software package of the author ([3]). The <italic>word-hyperbolic</italic> groups form an important subclass of the automatic groups. They are defined by the property that geodesic triangles in the Cayley graph of the group are uniformly thin. In other words, there is a constant <italic>ε</italic>, such a that any point on one of the sides of such a triangle is within distance <italic>ε</italic> of the union of the other two sides of the triangle. Small cancellation groups and groups acting discretely and cocompactly on hyperbolic space are examples of word-hyperbolic groups. Their theory is developed in [1]. The word problem in a word-hyperbolic group can be solved by a <italic>Dehn algorithm</italic> in linear time. Furthermore, Epstein has shown that the conjugacy problem can be solved in time <italic>O</italic>(<italic>n</italic> ln <italic>n</italic>). However, both of these algorithms require a prior knowledge of the constant <italic>ε</italic>, and it does not seem to be easy to compute this constant in general. In practice, the quadratic time algorithm for the word problem coming from the automatic groups procedures performs very effectively, and it would be useful to have alternatives for the conjugacy problem. The automatic groups machinery does provide such an algorithm (which works in the the intermediate class of <italic>biautomatic groups</italic>), but that unfortunately is exponential. Alternative algorithms for conjugacy testing of elements and subgroups are currently being developed and implemented by a research student of mine, Joe Marshall. They have the minor disadvanatage that their basic versions work only in torsion-free groups, but adaptations to the more general case seem possible. They include also a test for <italic>malnormality</italic> of a subgroup <italic>H</italic> of <italic>G</italic>, which means that <italic>g</italic><supscrpt>-1</supscrpt> <italic>Hg</italic>∩<italic>H</italic> = 1 for all <italic>g</italic> ∈ <italic>G</italic>\<italic>H</italic>. This test is important in topological applications. They make essential use of the <italic>boundary</italic> of the group. The points on the boundary can be defined as the equivalence classes of infinite geodesic rays starting from the base point of the Cayley graph, where two such rays are equivalent if they remain a bounded distance apart. The precise bound involved here turns out to be another universal constant <italic>δ</italic> of the Cayley graph, but unlike <italic>ε</italic> it can be computed in practice relatively easily. Roughly speaking, a certain linear function of <italic>δ</italic> provides us with a bound on the length of the elements <italic>g</italic> that we need to consider as possible conjugating elements when testing for conjugacy or malnormality. To make the algorithm practical, we need to restrict this set of potential conjugators considerably more than this, and Marshall has come up with various tricks for doing this.