Research Papers

Deciding if the Angle is Zero Inside Free Groups (19 pages)

This paper answers a question of Stallings about how to tell if the angle between finitely generated subgroups of a free group is zero. In the first section I define the angle and give some examples with example 1.9 showing how weird things can be. This example answers another question of Stallings negatively by giving subgroups whose angle cannot be measured using a geometric method invented by Stallings. In the second section I show how to decide if the angle is zero using a graph based method; the hi-light of this section is algorithm 2.9 which gives a method analogous to Gaussian elimination, but for free groups.

A slightly modified version of this paper appeared in Topology and its Applications, vol. 110, pp. 55-69, Elsevier Science B.V., Amsterdam, 2001.

Computing Angles In Hyperbolic Groups (31 pages)

This paper generalizes the results of the previous paper from free groups to hyperbolic groups. Because hyperbolic groups are a much more interesting bunch of groups, it's necessary to put some additional restrictions on the subgroups. My main theorem (6.3) is that it is possible to compute the angle between quasiconvex subgroups whose join (i.e. the subgroup generated by the union of the two groups) is also quasiconvex. There are groups ---such as the fundamental groups of hyperbolic surfaces--- in which all finitely generated subgroups are quasiconvex so this gives an algorithm for computing angles between all finitely generated subgroups of such groups. The proof of the main theorem depends on proving that the product of two geodesically regular subsets is again geodesically regular. This result is a little intricate and requires a construction I call the "balloon algorithm" which is a generalization of an algorithm due to R. Gilman. The analogous result that the product of two quasiconvex subsets of a hyperbolic group is again quasiconvex (proposition 3.14) is much easier to prove (in fact, the picture at the top of the page is the proof) and I wonder if a generalization of a theorem by Gersten and Short can be proved (question 4.8) which will give the result on geodesically regular subsets from the result on quasiconvex subsets.

This paper appears in pages 59 -- 88 of Groups, Languages and Geometry, R. Gilman Ed., Contemporary Mathematics, no. 250, American Mathematical Society, Providence, R.I., 1999.

Here's the table of contents:

1. Introduction
2. Rational Subsets
3. Quasiconvex Subsets
4. Geodesically Regular Subsets
5. The Angle Between Two Subgroups
6. Computing Positive Angles
7. Deciding if the Angle is Zero
8. The Ballon Construction

Given a hyperbolic group G, three collections of subsets are naturally definable. The first two classes ---rational subsets and quasiconvex subsets--- are classical. The third class ---regular subsets--- have often been called "rational"; to avoid ambiguity I reserve the term "rational" for the older concept, and use the term "regular" for Gersten and Short's newer concept. In sections 2 through 4 these three collections are defined, some basic examples are delineated, their closure properties are discussed, and the relationships amongst these collections is touched upon. In section 5 the group theoretic angle between subgroups is defined, and the reduction of angles to rational sets is explained; in addition, an example of a hyperbolic group in which angles are incomputable is given. The main theorem 6.3 is proved in section 6 modulo deciding if the angle is zero and the balloon construction. In section 7 all attention is focused on the zero angle case, and it is shown how to decide if the angle is zero when the subgroups and their join are quasiconvex. Finally, section 8 describes the balloon construction which generalizes an algorithm due to R. Gilman.