By Bhartendu Harishchandra, J.G.M. Mars

ISBN-10: 3540042326

ISBN-13: 9783540042327

Publication via Harishchandra, Bhartendu

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**Example text**

Then G is one of the following ﬁve groups: A : the group of order-preserving permutations of Q; B : the group of order-preserving or order-reversing permutations of Q; C : the group of permutations preserving the circular order on the set of complex roots of unity; Cameron: Infinite permutation groups 21 D : the group of permutations preserving or reversing the circular order on the set of complex roots of unity; S : the symmetric group. This theorem has wider implications. It follows that any highly set-transitive group of countable degree is contained as a dense subgroup in one of these ﬁve groups.

1+k+···+k n−1 . Since the intersection of all level stabilizers in a group of tree automorphisms is trivial (an automorphism fixing all the levels fixes the whole tree) we have the following proposition. 2 Every group acting faithfully on a regular rooted tree is residually finite. The group Sk (Sk (Sk · · · (Sk Sk ) . . 1 is the automorphism group, denoted Aut(T[n]) of the finite rooted k-ary tree T[n] consisting of levels 0 through n in T . The group Aut(T ) is a pro-finite group. Indeed, it is the inverse limit of the sequence 1 ← Aut(T[1]) ← Aut(T[2] ) ← .

Figure 1. The ternary rooted tree of words over X that start in u is a subtree of T , denoted Tu , which is canonically isomorphic to the whole tree through the isomorphism φu defined as the composition φu = φx1 . . φxn (see Figure 2). In particular, each φx , x ∈ X, is a canonical tree isomorphism between the tree T and the tree Tx hanging below the vertex x on the first level of T . The boundary of the tree T , denoted ∂T , is an ultrametric space whose points are the infinite geodesic rays in T starting at the root.

### Automorphic Forms on Semisimple Lie Groups by Bhartendu Harishchandra, J.G.M. Mars

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