Automorphisms on a Lie group contracting modulo a compact subgroup and applications to semistable convolution semigroups

Let ( _t ) _t0 be a -semistable convolution semigroup of probability measures on a Lie groupG whose idempotent ₀ is the Haar measure on some compact subgroupK. Then all the measures ₁ are supported by theK-contraction groupC _K() of the topological automorphism ofG. We prove here the structure theoremC _K()=C()K, whereC() is the contraction group of . Then it turns out that it is sufficient to study semistable convolution semigroups on simply connected nilpotent Lie groups that have Lie algebras with a positive graduation.     

Tame and Wild Coordinates of

Let be a field of characteristic zero. We characterize coordinates and tame coordinates in , i.e. the images of respectively under all automorphisms and under the tame automorphisms of . We also construct a new large class of wild automorphisms of which maps to a concrete family of nice looking polynomials. We show that a subclass of this class is stably tame, i.e. becomes tame when we extend its automorphisms to automorphisms of .

     

<Emphasis Type="Italic">G</Emphasis>-compactness and groups

Lascar described E _KP as a composition of E _L and the topological closure of E _L (Casanovas et al. in J Math Log 1(2):305–319). We generalize this result to some other pairs of equivalence relations. Motivated by an attempt to construct a new example of a non-G-compact theory, we consider the following example. Assume G is a group definable in a structure M. We define a structure M′ consisting of M and X as two sorts, where X is an affine copy of G and in M′ we have the structure of M and the action of G on X. We prove that the Lascar group of M′ is a semi-direct product of the Lascar group of M and G/G _L . We discuss the relationship between G-compactness of M and M′. This example may yield new examples of non-G-compact theories. The first author is supported by the Polish Goverment grant N N201 384134. The second author is supported by the Polish Goverment grant N201 032 32/2231.     

On automorphism groups of quasiprimitive 2-arc transitive graphs

We characterize the automorphism groups of quasiprimitive 2-arc-transitive graphs of twisted wreath product type. This is a partial solution for a problem of Praeger regarding quasiprimitive 2-arc transitive graphs. The solution stimulates several further research problems regarding automorphism groups of edge-transitive Cayley graphs and digraphs. This work forms part of an ARC grant project and is supported by a QEII Fellowship.     

Unimodal betti numbers for a class of nilpotent lie algebras

In this paper we prove that all finite dimens~onal complex nilpotent Lie algebras containing an abelian ideal of codimension 1 have unirnodal Betti numbers.     

Automorphisms of Models of True Arithmetic: Subgroups which Extend to a Maximal Subgroup Uniquely

We show that if M is a countable recursively saturated model of True Arithmetic, then G = Aut(M) has nonmaximal open subgroups with unique extension to a maximal subgroup of Aut(M). Mathematics Subject Classification: 03C62, 03C50.     

The Automaton Realization of Iterated Wreath Products of Cyclic Groups

We study profinite groups which are infinitely iterated wreath products W _∞ = …?C _{n 2} ?C _{n 1} of finite cyclic groups via combinatorial language of transducers. Namely, we provide a naturally defined automaton realization of the group W _∞ by an automaton over a changing alphabet. Our construction gives a characterization of these profinite groups as automaton groups, i.e. as groups generated by a single automaton.     

Multiplicative Automorphisms of Incidence Algebras

We present necessary and sufficient conditions for a multiplicative automorphism of an incidence algebra of a finite partially ordered set X to be an inner automorphism. These conditions are related to the structure of X as a directed graph.