By D.J. Collins, R.I. Grigorchuk, P.F. Kurchanov, H. Zieschang, A.I. Kostrikin, I.R. Shafarevich, P.M. Cohn

From the studies: "... The e-book below overview comprises monographs on geometric points of staff conception ... jointly, those articles shape a wide-ranging survey of combinatorial crew thought, with emphasis greatly at the geometric roots of the topic. this may be an invaluable reference paintings for the professional, in addition to supplying an outline of the topic for the outsider or amateur. many alternative subject matters are defined and explored, with the most effects provided yet now not proved. this permits the reader to get the flavor of those subject matters with no changing into slowed down intimately. either articles provide finished bibliographies, in order that it really is attainable to exploit this ebook because the start line for a extra specified research of a selected subject of curiosity. ..." Bulletin of the London Mathematical Society, 1996

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**Example text**

Let (U,H) denote a continuous,unitary, linear representation of G in the complex Hilbert space H. For k E ™u {+ let H-k be the strong topoiogicai antidual of the locally convex topological vector space Hk of k-times differentiable vectors for (U,H); cf. 2, Remark 3. Then we get the ascending filtration of continuous linear injections 00 } I o _co where H = H and the natural continuous sesquiiinear form <· ·> on H x H extends the restriction of the scalar product of H onto H x H • For every ~ E D(G) the t-linear mapping 00 00 00 U1(~) : H-oo ~Hoo is continuous with respect to the inductive limit topology of H- 00 and the projective limit topology of H = n Hk.

Indeed, for any element h EH the right translation by h- 1 EH in G defined according to is a homeomorphism of G into itself, we have 06 (1 6 ) = id 6 , o6 (hk) = o6 (h) o6 (k) for all pairs (h,k) E H x H, and the mapping defined via the prescription H x G 3 (h,x) ~ oG(h)x € 0 G is continuous with respect to the product topology of the direct product group H x G and the given group topology of G. H of H in G. , with the quotient by Hof the topology of G, then the locally compact topological space G/H is said to be the homogeneous space of left cosets of H in G.

G for y E G. 2. The unitarily induced linear representation (U,H) = Ind~(U 0 ,H0 ) of G is topologically irreducible only if (U ,H ) is a topologically irreducible 0 0 continuous, unitary, linear representa-don of H. Indeed, if (Ua,Hcx) are continuous, unitary, 1 i near representat i 011s of G for ex E { 1 ,2}, then Ind~(u 1 ~ u2 , H1 @ H2 ) and Ind~(u 1 ,H 1 )@ Ind~(u 2 ,H 2 ) are isomorphic, continuous, unitary, linear representations of G (cf. 4). Let Xu denote the central 0 character of ( U0 ,H0 ) (see 1.