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By Tammo tom Dieck

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Extra resources for Algebraic Topology and Tranformation Groups

Sample text

Moreover, it is easy to check that (1, ν) normalizes E. 5 ν is given by ⎛ 2 ⎜1 ⎜ ⎜ ⎜0 A=⎜ ⎜0 ⎜ ⎝0 0 3 0 0 0 2 0 0 0 0 2 −3 0 0 −1 2 0 0 0 0 7 0 0 0 4 ⎞ 0 0⎟ ⎟ ⎟ 0⎟ ⎟. 0⎟ ⎟ 12⎠ 7 √ √ √ √ Since the eigenvalues of A are 2 − 3, 2 + 3, 7 − 4 3, 7 + 4 3, the first two with multiplicity two, ν defines an Anosov diffeomorphism on M. 10 Let E be the group presented by E :=< a, b, c, d, e, f, α|[b, a] = d 2 , [c, a] = e2 , [c, b] = f 2 , αa = a −1 α, αb = b−1 α, αc = c−1 α, αd = dα αe = eα, α f = f α, α 2 = d > .

If is r -step nilpotent, the sequence r +1 =1⊂ r = cr ( ) ⊂ r −1 = c2 ( ) ⊂ 1 forms a central series with Fix now i/ i+1 = = cr −1 ( ) ⊂ · · · ⊂ 2 c1 ( ) = ∼ = Zki for some ki ∈ N. ar,1 , . . , ar,kr , ar −1,1 , . . , ar −1,kr−1 , . . , a2,1 , . . , a2,k2 , a1,1 , . . , a1,k1 , a set of generators for such that for any integer 1 ≤ i ≤ r , the classes a¯ i,1 , . . , a¯ i,ki ∈ i / i+1 freely generate the free abelian group i / i+1 . Any γ ∈ can be written as a product: v v v vr−1,k v v2,k v v1,k r,1 r−1,1 2,1 1,1 r−1 .

V2,1 , . . , v2,k2 , v1,1 , . . , v1,k1 ), which is a vector in Zk1 +···+kr . The notation γ (v) shows the dependence of γ on v. By [107] it is known that the product in is given by polynomial functions in v, that is, there exists a polynomial P : Z2(k1 +···+kr ) → Zk1 +···+kr such that γ (v1 )γ (v2 ) = γ (P(v1 , v2 )), for all v1 , v2 ∈ Z k1 +···+kr . The Malcev completion R is defined to be the set of all formal products 44 Principal classes of algebraic actions w w w wr−1,k w w2,k w w1,k r−1,1 r−1 n(w) = ar,1r,1 .

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Algebraic Topology and Tranformation Groups by Tammo tom Dieck


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