О многообразиях представлений некоторых свободных произведений циклических групп с одним соотношением

Abstract

In the paper representation varieties of two classes of finitely generated groups are investigated. The first class consists of groups with the presentation G = 〈a1, . . ., as, b1, . . ., bk, x1, . . ., xg | am11 = . . . = amsS = x21 . . . x2gW(a1, . . ., as, b1, . . ., bk) = 1〉, where g > 3, mi > 2 for i = 1, . . ., s and W(a1, . . ., as, b1, . . ., bk) is an element in normal form in the free product of cyclic groups H = 〈a1 | am11 〉 * . . . * 〈as | amsS 〉 * 〈b1〉 * . . . * 〈bk〉. The second class consists of groups with the presentation G(p, q) = 〈a1, . . ., as, b1, . . ., bk, x1, . . ., xg, t | am11 = . . . = amsS = 1, tUpt−1 = Uq〉, where p and q are integer numbers such that p > |q| ≥ 1, (p, q) = 1, mi > 2 for i = 1, . . ., s, g > 3, U = x21 . . . x2gW(a1, . . ., as, b1, . . ., bk) and W(a1, . . ., as, b1, . . ., bk) is an above defined element. Irreducible components of representation varieties Rn(G) and Rn(G(p, q)) are found, their dimensions are calculated and it is proved, that every irreducible component is a rational variety.