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Fully Irreducible Automorphism
In the mathematical subject geometric group theory, a fully irreducible automorphism of the free group ''Fn'' is an element of Out(Fn), Out(''Fn'') which has no periodic conjugacy classes of proper free factors in ''Fn'' (where ''n'' > 1). Fully irreducible automorphisms are also referred to as "irreducible with irreducible powers" or "iwip" automorphisms. The notion of being fully irreducible provides a key Out(''Fn'') counterpart of the notion of a pseudo-Anosov map, pseudo-Anosov element of the mapping class group of a finite type surface. Fully irreducibles play an important role in the study of structural properties of individual elements and of subgroups of Out(''Fn''). Formal definition Let \varphi\in \operatorname(F_n) where n\ge 2. Then \varphi is called ''fully irreducible'' if there do not exist an integer p\ne 0 and a proper free factor A of F_n such that \varphi^p([A])=[A], where [A] is the conjugacy class of A in F_n. Here saying that A is a proper free ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Group Theory
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such group (mathematics), groups and topology, topological and geometry, geometric properties of spaces on which these groups Group action (mathematics), act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces). Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph (discrete mathematics), graph structure, are endowed with the structure of a metric space, given by the so-called word metric. Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimens ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
