Aspects of Growth in Infinite Groups

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We will study various kinds of growth in infinite groups, with particular focus on the growth of conjugacy classes. Benson's 1983 proof that the standard growth series of a virtually abelian group is rational with respect to any generating set introduced the idea of partitioning the language of words over the generators into 'patterned sets', which allows a linear algebraic approach to growth. The same paper also provides a criterion for growth to be rational. We hope to make use of these ideas to investigate the growth of conjugacy classes in virtually abelian groups. Other areas of investigation include growth of cosets, and relative subgroup growth.