arXiv subjects

Alexei Myasnikov

Publications and source records attributed to Alexei Myasnikov.

41 records · Page 3Linked to original sources

arXiv2005

Algebraic Geometry over Free Groups: Lifting Solutions into Generic Points

In this paper we prove Implicit Function Theorems (IFT) for algebraic varieties defined by regular quadratic equations and, more generally, regular NTQ systems over free groups. In the model theoretic language these results state the existence of very simple Skolem functions for particular $\forall\exists$-formulas over free groups. We construct these functions effectively. In non-effective form IFT first appeared in \cite{Imp}. From algebraic geometry view-point IFT can be described as lifting solutions of equations into generic points of algebraic varieties. Moreover, we show that the converse is also true, i.e., IFT holds only for algebraic varieties defined by regular NTQ systems. This implies that if a finitely generated group $H$ is $\forall\exists$-equivalent to a free non-abelian group then $H$ is isomorphic to the coordinate group of a regular NTQ system.

math.GR↗

arXiv2003

Foldings, graphs of groups and the membership problem

We introduce a combinatorial version of Stallings-Bestvina-Feighn-Dunwoody folding sequences. We then show how they are useful in analyzing the solvability of the uniform subgroup membership problem for fundamental groups of graphs of groups. Applications include coherent right-angled Artin groups and coherent solvable groups.

math.GR↗

arXiv2002

Average-case complexity and decision problems in group theory

We investigate the average-case complexity of decision problems for finitely generated groups, in particular the word and membership problems. Using our recent results on ``generic-case complexity'' we show that if a finitely generated group $G$ has the word problem solvable in subexponential time and has a subgroup of finite index which possesses a non-elementary word-hyperbolic quotient group, then the average-case complexity of the word problem for $G$ is linear time, uniformly with respect to the collection of all length-invariant measures on $G$. For example, the result applies to all braid groups $B_n$.

math.GR↗

arXiv2002

Generic-case complexity, decision problems in group theory and random walks

We give a precise definition of ``generic-case complexity'' and show that for a very large class of finitely generated groups the classical decision problems of group theory - the word, conjugacy and membership problems - all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs.

math.GR↗

arXiv2002

Stallings foldings and the subgroup structure of free groups

We re-cast in a more combinatorial and computational form the foldings approach of John Stallings and pursue a detailed study of the subgroup structure of free groups. In particular, we introduce the notions of an "algebraic" and a "free" extension for subgroups of free groups and apply these notions to obtain a number of algebraic and algorithmic results.

math.GR↗