On automorphism groups of connected Lie groups

We prove that ifG is a connected Lie group with no compact central subgroup of positive dimension then the automorphism group ofG is an almost algebraic subgroup of , where is the Lie algebra ofG. We also give another proof of a theorem of D. Wigner, on the connected component of the identity in the automorphism group of a general connected Lie group being almost algebraic, and strengthen a result of M.Goto on the subgroup consisting of all automorphisms fixing a given central element.     

Finite almost simple groups with prime graphs all of whose connected components are cliques

We find finite almost simple groups with prime graphs all of whose connected components are cliques, i.e., complete graphs. The proof is based on the following fact, which was obtained by the authors and is of independent interest: the prime graph of a finite simple nonabelian group contains two nonadjacent odd vertices that do not divide the order of the outer automorphism group of this group.     

Growth of connected locally compact groups

The asymptotic behavior of the Haar measure of Uⁿ = U · U ··· U, where U is a compact neighborhood of the identity in a separable, connected locally compact group, is considered. It is shown that for a given group G, the measure of Uⁿ grows at most polynomially for all U ? G, or at least exponentially for all U ? G. The groups having polynomial growth are characterized in terms of uniformly discrete free subsemigroups and in terms of the eigenvalues of elements occurring in the adjoint group of an approximating Lie group.     

On open subsemigroups of connected groups

It is well known that a cancellative semigroup can be embedded into a group if it satisfies “Ore’s condition” of being either left or right reversible. However Ore’s condition is by no means necessary, so it is natural to ask which subsemigroups of a group are left or right reversible, or satisfy a condition of a similar type. In the present paper we study this question on open subsemigroups of connected locally compact groups; we also show how to use concepts related with reversibility to prove assertions like the following: Suppose thatS is an open subsemigroup of a connected Lie groupG such that 1 ∈ . IfG is solvable or ifS is invariant thenS is connected andS determinesG uniquely; that is to say, ifS can be embedded as an open subsemigroup into a connected Lie groupG’ thenG’ is isomorphic withG. Examples show that there are non-connected open subsemigroupsS of Sl(2,R) with 1 ∈ and such that the uniqueness assertion fails. The author gratefully acknowledges the support he received from the Alexander von Humboldt Foundation during the time he prepared this paper.     

Realizing connected Lie groups as automorphism groups of complex manifolds

We show that every connected real Lie group can be realized as the full automorphism group of a Stein hyperbolic complex manifold.     

The Kuratowski convergence and connected components

We investigate the Kuratowski convergence of the connected components of the sections of a definable set applying the result obtained to semialgebraic approximation of subanalytic sets. We are led to some considerations concerning the connectedness of the limit set in general. We discuss also the behaviour of the dimension of converging sections and prove some general facts about the Kuratowski convergence in tame geometry.     

Twisted conjugacy in simply connected groups

If G is a group and an automorphism of G, one has the twisted conjugation action of G on itself This paper collects a number of results — more or less well known — for the case that G is a simply connected semisimple group.     

Model theoretic dynamics in Galois fashion

We fix a monster model $\mathfrak{D}$ of some stable theory and investigate substructures of $\mathfrak{D}$ which are existentially closed within the class of substructures equipped with an action of a fixed group G. We describe them as PAC substructures of $\mathfrak{D}$ and obtain results related to Galois theory.Assuming that the class of these existentially closed substructures is elementary, we show that, under the assumption of having bounded models, its theory is simple and eliminates quantifiers up to some existential formulas. Moreover, this theory codes finite sets and allows a geometric elimination of imaginaries, but not always a weak elimination of imaginaries.     

Partial isometries: Factorization and connected components

In this paper we compute the distance between the connected components of the sets of all partial isometries and essential partial isometries. Some factorization results are also proved. This complements previous results due to P.R. Halmos and J.E. McLaughlin