TY - JOUR

T1 - Universal Equivalence of Generalized Baumslag–Solitar Groups

AU - Dudkin, F. A.

N1 - Funding Information: Supported by SB RAS Fundamental Research Program I.1.1, project No. 0314-2019-0001. Publisher Copyright: © 2020, Springer Science+Business Media, LLC, part of Springer Nature. Copyright: Copyright 2020 Elsevier B.V., All rights reserved.

PY - 2020/11/1

Y1 - 2020/11/1

N2 - A finitely generated group acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (a GBS group). Every GBS group is the fundamental group Π1(A) of a suitable labeled graph A. We prove that if A and B are labeled trees, then the groups Π1(A) and Π1(B) are universally equivalent iff Π1(A) and Π1(B) are embeddable into each other. An algorithm for verifying universal equivalence is pointed out. Moreover, we specify simple conditions for checking this criterion in the case where the centralizer dimension is equal to 3.

AB - A finitely generated group acting on a tree so that all vertex and edge stabilizers are infinite cyclic groups is called a generalized Baumslag–Solitar group (a GBS group). Every GBS group is the fundamental group Π1(A) of a suitable labeled graph A. We prove that if A and B are labeled trees, then the groups Π1(A) and Π1(B) are universally equivalent iff Π1(A) and Π1(B) are embeddable into each other. An algorithm for verifying universal equivalence is pointed out. Moreover, we specify simple conditions for checking this criterion in the case where the centralizer dimension is equal to 3.

KW - embedding of groups

KW - existential equivalence

KW - generalized Baumslag–Solitar group

KW - universal equivalence

UR - http://www.scopus.com/inward/record.url?scp=85096847064&partnerID=8YFLogxK

U2 - 10.1007/s10469-020-09609-5

DO - 10.1007/s10469-020-09609-5

M3 - Article

AN - SCOPUS:85096847064

VL - 59

SP - 357

EP - 366

JO - Algebra and Logic

JF - Algebra and Logic

SN - 0002-5232

IS - 5

ER -