On the Generalized Second Nilpotent Product of Groups

We introduce the notion of(λ,μ)-product ofL-subsets. We give a necessary and sufficient condition for(λ,μ)-L-subgroup of a product of groups to be(λ,μ)-product of(λ,μ)-L-subgroups.

Download Full-text

A NOTE ON THE NORMS OF TRANSITION OPERATORS ON LAMPLIGHTER GRAPHS AND GROUPS

International Journal of Algebra and Computation ◽

10.1142/s0218196705002591 ◽

2005 ◽

Vol 15 (05n06) ◽

pp. 1261-1272 ◽

Cited By ~ 6

Author(s):

WOLFGANG WOESS

Keyword(s):

Random Walk ◽

Wreath Products ◽

Locally Compact Group ◽

Operator Matrix ◽

Locally Compact ◽

Finitely Generated Groups ◽

Group Of Isometries ◽

Transition Operators ◽

Special Case ◽

Product Of Groups

Let L≀X be a lamplighter graph, i.e., the graph-analogue of a wreath product of groups, and let P be the transition operator (matrix) of a random walk on that structure. We explain how methods developed by Saloff-Coste and the author can be applied for determining the ℓp-norms and spectral radii of P, if one has an amenable (not necessarily discrete or unimodular) locally compact group of isometries that acts transitively on L. This applies, in particular, to wreath products K≀G of finitely-generated groups, where K is amenable. As a special case, this comprises a result of Żuk regarding the ℓ2-spectral radius of symmetric random walks on such groups.

Download Full-text

UNDECIDABILITY OF THE THEORIES OF CLASSES OF STRUCTURES

Journal of Symbolic Logic ◽

10.1017/jsl.2014.54 ◽

2014 ◽

Vol 79 (4) ◽

pp. 1001-1019 ◽

Cited By ~ 2

Author(s):

ASHER M. KACH ◽

ANTONIO MONTALBÁN

Keyword(s):

Direct Product ◽

Disjoint Union ◽

Second Order ◽

Linear Orders ◽

Second Order Arithmetic ◽

Order Arithmetic ◽

Natural Classes ◽

Product Of Groups

AbstractMany classes of structures have natural functions and relations on them: concatenation of linear orders, direct product of groups, disjoint union of equivalence structures, and so on. Here, we study the (un)decidability of the theory of several natural classes of structures with appropriate functions and relations. For some of these classes of structures, the resulting theory is decidable; for some of these classes of structures, the resulting theory is bi-interpretable with second-order arithmetic.

Download Full-text