Symmetry And Group

A 2-generated just-infinite profinite group which is not by Lucchhini A.

By Lucchhini A.

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Then d exp0 : T0 g → Te G. But T0 g can be identified with g and Te G = g. Hence d exp0 linearly maps g to g. Proposition d exp0 = id. There is an open neighbourhood U of 0 in g and an open neighbourhood V of e in G such that exp: U → V is a C ∞ diffeomorphism. Then the inverse map exp−1 : V → U is denoted by log. Proof The second statement follows by the inverse function theorem. For the proof of the first statement let A ∈ g, α(t) := tA and β(t) := exp(tA). Thus α represents A ∈ T0 g and A = β ′ (0) = d exp0 (A).

O(|t|n+1 ) as t → 0. 22 have Proposition Let G and g be as above. f ). Proof Let x ∈ G. We will expand f (x exp(tA) exp(tB) exp(−tA) exp(−tB)) in two different ways as a Taylor series in t up to degree 2, where t → 0 in R. Then we obtain the result by equality of second degree terms in both expansions. f ))(x) + O(|t|3 ). f )(x) + O(|t|3 ). 23 Corollary Let G be a Lie group with g := Te G and Lie(G) the Lie algebra of left invariant vector fields on G. Put [A, B] := b(A, B) (A, B ∈ g). f . Ex.

F ) . )))(x) k 1 ! . km ! + O(|t|n+1 ) as t → 0. 22 have Proposition Let G and g be as above. f ). Proof Let x ∈ G. We will expand f (x exp(tA) exp(tB) exp(−tA) exp(−tB)) in two different ways as a Taylor series in t up to degree 2, where t → 0 in R. Then we obtain the result by equality of second degree terms in both expansions. f ))(x) + O(|t|3 ). f )(x) + O(|t|3 ). 23 Corollary Let G be a Lie group with g := Te G and Lie(G) the Lie algebra of left invariant vector fields on G. Put [A, B] := b(A, B) (A, B ∈ g).

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