Decision Problems in Group Theory

Abstract

We study finiteness problems for isogeny classes of abelian varieties over an algebraic function field K in one variable over the field of complex numbers. In particular, we construct explicitly a non‐isotrivial absolutely simple abelian fourfold X over a certain K such that the isogeny class of X×X contains infinitely many mutually non‐isomorphic principally polarized abelian varieties. (Such examples do not exist when the ground field is finitely generated over its prime subfield.)

Gröbner–Shirshov basis for the Braid group in the Birman–Ko–Lee generators

“…As the result, it gives simple analysis of Novikov and Boone groups and simple alternative proofs of the main results of that papers. In the same way, Kalorkoti [37,38] found in fact Gröbner-Shirshov bases of some groups using the above notion of group with standard basis. It gives simple proofs of Collins [32,33] and Bokut [10] results for Turing degrees of word and conjugacy problems for finitely presented groups.…”

A Finitely Presented Group With Almost Solvable Conjugacy Problem

“…As shown in [12], it is easy to see that E 1 is an HNN extension of E 0 and has solvable word problem; it is a straightforward matter to extend this to E 0 , . .…”

“…We will use a group introduced in §4 of [12] but with extra relations (those of E 2 and E 3 ); these take care of the awkward case of s-fold computation when gcd(m, s) = 1. Let the quadruples of the modular machine M of the preceding section be (a i , b i , c i , R) for i ∈ I and (a j , b j , c j , L) for j ∈ J.…”

“…Conversely if W is a positive { r i , l j }-word that satisfies the equation, then (α 0 , β 0 ) → s (α n , β n ). The following is a more general version of Lemma 1 of [12].…”

“…We show that E 3 has Bokut' normal forms [3]; see also [5] and [6] (a brief introduction is given in [12]). We set:…”