Lecture 17 : Grigorchuk ’ s co - growth criteria
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Growth and Amenability of the Grigorchuk Group
In this note, we discuss the amenability of the first Grigorchuk group by appealing to its growth properties. In particular, we show that the first Grigorchuk group Γ has intermediate growth, hence providing an exotic example of an amenable group with non-polynomial growth. We also note some other interesting properties of Γ without proof, as well as a few examples of important groups construct...
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We present an analytic technique for estimating the growth for groups of intermediate growth. We apply our technique to Grigorchuk groups, which are the only known examples of such groups. Our estimates generalize and improve various bounds by Grigorchuk, Bartholdi and others. 2000 Mathematics Subject Classification: 20E08, 20E69, 68R15
متن کاملGrowth in Product Replacement Graphs of Grigorchuk Groups
The product replacement graph Γk(G) is the graph on the generating k-tuples of a group G, with edges corresponding to Nielsen moves. We prove the exponential growth of product replacement graphs Γk(Gω) of Grigorchuk groups, for k ≥ 5.
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A theory for groups of diffeomorphisms of the interval has been extensively developed by many authors (see for example [17, 19, 20, 25, 27, 30, 31, 32, 33, 35]). One of the most interesting topics of this theory is the interplay between the differentiability class of the diffeomorphisms and the algebraic (as well as dynamical) properties of the group (and the action). For instance, as a consequ...
متن کاملOperator Growth Functions of Discrete Groups
A generalizationof the growth functions of nitelygeneratedgroups is introduced, namely the growth functionswith operatorcoeecients g2G gz jgj. The questions of rationality and convergence of such series are discussed. The operator growth functions of surface groups are explicitly computed. The operator geodesic growth functions are also studied.
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تاریخ انتشار 2014