Alexei G. Myasnikov

Algebraic Geometry over Groups

Classical commutative algebra provides the underpinnings of classical algebraic geometry. In this paper we will describe, without any proofs, a theory for groups that parallels this commutative algebra and that, in like fashion, is the basis of what we term algebraic geometry over groups.

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

Abstract 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.

EXPONENTIAL GROUPS 2: EXTENSIONS OF CENTRALIZERS AND TENSOR COMPLETION OF CSA-GROUPS

For a CSA group

FOLDINGS, GRAPHS OF GROUPS AND THE MEMBERSHIP PROBLEM

G

Non-Commutative Cryptography and Complexity of Group-Theoretic Problems

and a wide class of abelian groups

Unification theorems in algebraic geometry

A

MULTIPLICATIVE MEASURES ON FREE GROUPS

we give an explicit construction for the tensor

A practical attack on a braid group based cryptographic protocol

A

Tarski’s problem about the elementary theory of free groups has a positive solution

-completion of

Random subgroups and analysis of the length-based and quotient attacks

G