Abstract Given a root class documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$ {mathcal{C}} $end{document} of groups, we establish some necessary and sufficient conditions for the documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$ {mathcal{C}} $end{document}-residuality of the free product of the two groups with amalgamated subgroups one of which is central in the corresponding free factor. We apply these results to obtain the criteria and sufficient conditions for the documentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} begin{document}$ {mathcal{C}} $end{document}-residuality of some HNN-extensions and tree products.
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