Alexei Miasnikov

From automatic structures to automatic groups

In this paper we introduce the concept of a Cayley graph automatic group (CGA group or graph automatic group, for short) which generalizes the standard notion of an automatic group. Like the usual automatic groups graph automatic ones enjoy many nice properties: these group are invariant under the change of generators, they are closed under direct and free products, certain types of amalgamated products, and finite extensions. Furthermore, the Word Problem in graph automatic groups is decidable in quadratic time. However, the class of graph automatic groups is much wider then the class of automatic groups. For example, we prove that all finitely generated 2-nilpotent groups and Baumslag-Solitar groups B(1,n) are graph automatic, as well as many other metabelian groups.

Algebraic Extensions in Free Groups

The aim of this paper is to unify the points of view of three recent and independent papers (Ventura 1997, Margolis, Sapir and Weil 2001 and Kapovich and Miasnikov 2002), where similar modern versions of a 1951 theorem of Takahasi were given. We develop a theory of algebraic extensions for free groups, highlighting the analogies and differences with respect to the corresponding classical field-theoretic notions, and we discuss in detail the notion of algebraic closure. We apply that theory to the study and the computation of certain algebraic properties of subgroups (e.g., being malnormal, pure, inert or compressed, being closed in certain profinite topologies) and the corresponding closure operators. We also analyze the closure of a subgroup under the addition of solutions of certain sets of equations.

A hierarchical projection pursuit clustering algorithm

We define a cluster to be characterized by regions of high density separated by regions that are sparse. By observing the downward closure property of density, the search for interesting structure in a high dimensional space can be reduced to a search for structure in lower dimensional subspaces. We present a hierarchical projection pursuit clustering (HPPC) algorithm that repeatedly bi-partitions the dataset based on the discovered properties of interesting 1-dimensional projections. We describe a projection search procedure and a projection pursuit index function based on Cho, Haralick and Yis improvement of the Kittler and Illingworth optimal threshold technique. The output of the algorithm is a decision tree whose nodes store a projection and threshold and whose leaves represent the clusters (classes). Experiments with various real and synthetic datasets show the effectiveness of the approach.

Pattern recognition and minimal words in free groups of rank 2

Abstract We describe a linear time probabilistic algorithm to recognize Whitehead minimal elements (elements of minimal length in their automorphic orbits) in free groups of rank 2. For a non-minimal element the algorithm gives an automorphism that is most likely to reduce the length of the element. This method is based on linear regression and pattern recognition techniques.

Random nilpotent groups, polycyclic presentations, and Diophantine problems

Abstract We introduce a model of random finitely generated, torsion-free, 2-step nilpotent groups (in short, τ 2 {\tau_{2}} -groups). To do so, we show that these are precisely the groups with presentation of the form 〈 A , C ∣ [ a i , a j ] = ∏ t = 1 m c t λ t , i , j ( 1 ≤ i < j ≤ n ) , [ A , C ] = [ C , C ] = 1 〉 {\langle A,C\mid[a_{i},a_{j}]=\prod_{t=1}^{m}c_{t}^{\lambda_{t,i,j}}(1\leq i<j% \leq n),\,[A,C]=[C,C]=1\rangle} , where A = { a 1 , … , a n } {A=\{a_{1},\dots,a_{n}\}} and C = { c 1 , … , c m } {C=\{c_{1},\dots,c_{m}\}} . Hence, a random G can be selected by fixing A and C, and then randomly choosing integers λ t , i , j {\lambda_{t,i,j}} , with | λ t , i , j | ≤ ℓ {|\lambda_{t,i,j}|\leq\ell} for some ℓ {\ell} . We prove that if m ≥ n - 1 ≥ 1 {m\geq n-1\geq 1} , then the following hold asymptotically almost surely as ℓ → ∞ {\ell\to\infty} : the ring ℤ {\mathbb{Z}} is e-definable in G, the Diophantine problem over G is undecidable, the maximal ring of scalars of G is ℤ {\mathbb{Z}} , G is indecomposable as a direct product of non-abelian groups, and Z ⁢ ( G ) = 〈 C 〉 {Z(G)=\langle C\rangle} . We further study when Z ⁢ ( G ) ≤ Is ⁡ ( G ′ ) {Z(G)\leq\operatorname{Is}(G^{\prime})} . Finally, we introduce similar models of random polycyclic groups and random f.g. nilpotent groups of any nilpotency step, possibly with torsion. We quickly see, however, that the latter yields finite groups a.a.s.