arXiv subjects

Alexei G. Myasnikov

Publications and source records attributed to Alexei G. Myasnikov.

25 records · Page 2Linked to original sources

arXiv2003

Whitehead method and Genetic Algorithms

In this paper we discuss a genetic version (GWA) of the Whitehead's algorithm, which is one of the basic algorithms in combinatorial group theory. It turns out that GWA is surprisingly fast and outperforms the standard Whitehead's algorithm in free groups of rank >= 5. Experimenting with GWA we collected an interesting numerical data that clarifies the time-complexity of the Whitehead's Problem in general. These experiments led us to several mathematical conjectures. If confirmed they will shed light on hidden mechanisms of Whitehead Method and geometry of automorphic orbits in free groups.

math.GR↗

arXiv2003

On the Andrews-Curtis equivalence

The Andrews-Curtis conjecture claims that every balanced presentation of the trivial group can be reduced to the standard one by a sequence of ``elementary transformations" which are Nielsen transformations augmented by arbitrary conjugations. It is a prevalent opinion that this conjecture is false; however, not many potential counterexamples are known. In this paper, we show that some of the previously proposed examples are actually not counterexamples. We hope that the tricks we used in constructing relevant chains of elementary transformations will be useful to those who attempt to establish the Andrews-Curtis equivalence in other situations. On the other hand, we give two rather general and simple methods for constructing balanced presentations of the trivial group; some of these presentations can be considered potential counterexamples to the Andrews-Curtis conjecture. One of the methods is based on a simple combinatorial idea of composition of group presentations, whereas the other one uses "exotic" knot diagrams of the unknot. We also consider the Andrews-Curtis equivalence in metabelian groups and reveal some interesting connections of relevant problems to well-known problems in K-theory.

math.GR↗

arXiv2003

On the Hanna Neumann Conjecture

The Hanna Neumann conjecture states that if F is a free group, then for all nontrivial finitely generated subgroups H,K <= F, rank(H intersect K) - 1 <= [rank(H)-1] [rank(K)-1]. Where most papers to date have considered a direct graph theoretic interpretation of the conjecture, here we consider the use of monomorphisms. We illustrate the effectiveness of this approach with two results. First, we show that for any finitely generated groups H,K <= F either the pair H,K or the pair H^{-}, K satisfy the Hanna Neumann conjecture--Here {-} denotes the automorphism which sends each generator of F to its inverse. Next, using particular monomorphisms from F to F_2, we obtain that if the Hanna Neumann conjecture is false then there is a counterexample H,K < F_2 having the additional property that all the branch vertices in the foldings of H and K are of degree 3, and all degree 3 vertices have the same local structure or ``type''.

math.GR↗

arXiv2002

Automorphic orbits in free groups

Let $F_n$ be the free group of a finite rank $n$. We study orbits $Orb_ϕ(u)$, where $u$ is an element of the group $F_n$, under the action of an automorphism $ϕ$. If an orbit like that is finite, we determine precisely what its cardinality can be if $u$ runs through the whole group $F_n$, and $ϕ$ runs through the whole group $Aut(F_n)$. Another problem that we address here is related to Whitehead's algorithm that determines whether or not a given element of a free group of finite rank is an automorphic image of another given element. It is known that the first part of this algorithm (reducing a given free word to a free word of minimum possible length by elementary Whitehead automorphisms) is fast (of quadratic time with respect to the length of the word). On the other hand, the second part of the algorithm (applied to two words of the same minimum length) was always considered very slow. We give here an improved algorithm for the second part, and we believe this algorithm always terminates in polynomial time with respect to the length of the words. We prove that this is indeed the case if the free group has rank 2.

math.GR↗

arXiv2002

Multiplicative measures on free groups

We introduce a family of atomic measures on free groups generated by no-return random walks. These measures are shown to be very convenient for comparing "relative sizes" of subgroups, context-free and regular subsets (that, subsets generated by finite automata) of free groups. Many asymptotic characteristics of subsets and subgroups are naturally expressed as analytic properties of related generating functions. We introduce an hierarchy of asymptotic behaviour "at infinity" of subsets in the free groups, more sensitive than the traditionally used asymptotic density, and apply it to normal subgroups and regular subsets.

math.GR↗

arXiv2002

Measuring sets in infinite groups

We are now witnessing a rapid growth of a new part of group theory which has become known as "statistical group theory". A typical result in this area would say something like ``a random element (or a tuple of elements) of a group G has a property P with probability p". The validity of a statement like that does, of course, heavily depend on how one defines probability on groups, or, equivalently, how one measures sets in a group (in particular, in a free group). We hope that new approaches to defining probabilities on groups outlined in this paper create, among other things, an appropriate framework for the study of the "average case" complexity of algorithms on groups.

math.GR↗

arXiv2001

The Andrews-Curtis Conjecture and Black Box Groups

The paper discusses the Andrews-Curtis graph of a normal subgroup N in a group G. The vertices of the graph are k-tuples of elements in N which generate N as a normal subgroup; two vertices are connected if one them can be obtained from another by certain elementary transformations. This object appears naturally in the theory of black box finite groups and in the Andrews-Curtis conjecture in algebraic topology. We suggest an approach to the Andrews-Curtis conjecture based on the study of Andrews-Curtis graphs of finite groups, discuss properties of Andrews-Curtis graphs of some classes of finite groups and results of computer experiments with generation of random elements of finite groups by random walks on their Andrews-Curtis graphs.

math.GR↗