The automorphism groups of compact homogeneous spaces

Considering the automorphism groups of compact homogeneous spaces we inspect certain general properties, indicate a method for calculating the groups, and illustrate it with examples in a few particular cases.     

Banach spaces in which Dunford-Pettis sets are relatively compact

Work performed under the auspices of G.N.A.F.A. of C.N.R. and partially supported by M.U.R.S.T. of Italy (40%)     

Compact groups having almost discrete orbit hypergroups

LetG be a compact group of automorphism acting continuously on a compact groupH. Then the orbit spaceH ^G is a compact hypergroup. We characterize, all solvable groupsH and compact automorphism groupsG for whichH ^G is almost discrete, i.e.,H ^G is homeomorphic to the one-point-compactification of . It turns out that thenH is isomorphic either to the infinite direct product (p) of the cyclic groups (p) or to _p ⁿ ( _p the group of allp-adic numbers) for some primep and some . The almost discrete orbit hypergroupsH ^G are determined explicitly for some examples.     

Small conjugacy classes in the automorphism groups of relatively free groups

Let F be an infinitely generated free group and let R be a fully invariant subgroup of F such that (a) R is contained in the commutator subgroup F^′ of F and (b) the quotient group F/R is residually torsion-free nilpotent. Then the automorphism group of the group F/R^′ is complete. In particular, the automorphism group of any infinitely generated free solvable group of derived length at least two is complete.This extends a result by Dyer and Formanek (1977) [7] on finitely generated groups F_n/R^′ where F_n is a free group of finite rank n at least two and R a characteristic subgroup of F_n.     

Metacyclic groups as automorphism groups of compact Riemann surfaces

Let X be a compact Riemann surface of genus \(g\ge 2\), and let G be a subgroup of \(\mathrm{Aut}(X)\). We show that if the Sylow 2-subgroups of G are cyclic, then \(|G|\le 30(g-1)\). If all Sylow subgroups of G are cyclic, then, with two exceptions, \(|G|\le 10(g-1)\). More generally, if G is metacyclic, then, with one exception, \(|G|\le 12(g-1)\). Each of these bounds is attained for infinitely many values of g.     

Banach spaces in which weakly p -Dunford–Pettis sets are relatively compact

In this paper we introduce p-Dunford–Pettis completely continuous operators and study Banach spaces with the wp-Dunford–Pettis relative compact property (wp-DPrcP). We study the behaviour of p-Dunford–Pettis completely operators on spaces with this property. We give sufficient conditions for spaces of operators to have the wp-DPrcP.     

Solvable line-transitive automorphism groups of finite linear spaces

Let S be a finite linear space, and letG be a group of automorphisms of S. IfG is soluble and line-transitive, then for a givenk but a finite number of pairs of (S, G),S hasv= p ⁿ points andG ⩽AΓL(1,p ⁿ ).     

A characterization of compact complex tori via automorphism groups

We show that a compact Kähler manifold $X$ is a complex torus if both the continuous part and discrete part of some automorphism group $G$ of $X$ are infinite groups, unless $X$ is bimeromorphic to a non-trivial $G$ -equivariant fibration. Some applications to dynamics are given.     

Hardy type spaces associated with compact unitary groups

We investigate Hilbertian Hardy type spaces of complex analytic functions of infinite many variables, associated with compact unitary groups and the corresponding invariant Haar’s measures. For such analytic functions we establish a Cauchy type integral formula and describe natural domains. Also we show some relations between constructed spaces of analytic functions and the symmetric Fock space.     

Lipschitz spaces on compact Lie groups

We prove the equivalence between Lipschitz spaces on a compact Lie group defined in terms of Weierstrass integrals and by means of higher order difference operators.