Papers on AC conjecture

1. Finitary Andrews-Curtis conjecture (with A.V. Borovik and A. Lubotzky) Sumbmitted.

We formulate a generalized AC-conjecture for an arbitrary group and prove that it holds for every finite group. In fact, there are no counter examples to the generalized AC conjecture so far. Such examples would shed some light on potential counterexamples to the original AC conjecture.

2. The Andrews-Curtis Conjecture and Black Box Groups (with A.V. Borovik and E.I. Khukhro) Internat. J. Algebra Comput, 13 (2003), no. 4, 415-436.

We introduce Andrews-Curtis graphs of groups and discuss their connection with black-box groups and the replacement algorithm, study AC conjecture for finite groups, and describe possible attacks on AC conjecture via finite groups.

3. On the Andrews-Curtis equivalence (with A.D. Miasnikov and V. Shpilrain). Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), 183-198. Contemp. Math., 296, Amer. Math. Soc., Providence, RI, 2002.

We construct various examples of balanced presentations of the trivial group and show that some of them satisfy AC conjecture. We also consider AC-equivalence in metabelian groups and reveal some interesting connections of it with well-known problems in K-theory.

4. Balanced presentations of the trivial group on two generators and the Andrews-Curtis conjecture (with A.D. Miasnikov). Groups and computation, III (Columbus, OH, 1999), 257-263. Ohio State Univ. Math. Res. Inst. Publ. 8, de Gruyter, Berlin, 2001.

We proved that every balanced presentation on two generators of the trivial group satisfies the AC conjecture.

5. Extended Nielsen transformations and the trivial group. (Russian) Mat. Zametki 35 (1984), no. 4, 491-495.

It is an old paper, where I showed that AC conjecture holds in free solvable groups.