Dead ends on wreath products and lamplighter groups

Dead End Words in Lamplighter Groups and Other Wreath Products Sean Cleary and Jennifer Taback

We explore the geometry of the Cayley graphs of the lamplighter groups and a wide range of wreath products. We show that these groups have dead end elements of arbitrary depth with respect to their natural generating sets. An element w in a group G with finite generating set X is a dead end element if no geodesic ray from the identity to w in the Cayley graph Γ(G, X) can be extended past w. Add...

Wreath Products and Existentially Complete Solvable Groups

It is known that the theory of abelian groups has a model companion but that the theory of groups does not. We show that for any fixed 22 a 2 the theory of groups solvable of length s 22 has no model companion. For the metabelian case (22 = 2) we prove the stronger result that the classes of finitely generic, infinitely generic, and existentially complete metabelian groups are all distinct. We ...

Two Problems on Varieties of Groups Generated by Wreath Products

Group Codes based on Wreath Products of Complex Reflection Groups

Group codes based on wreath products of complex matrix groups are constructed. Efficient algorithms for encoding and decoding these codes are described.

Some Remarks on Depth of Dead Ends in Groups

It is known, that the existence of dead ends (of arbitrary depth) in the Cayley graph of a group depends on the chosen set of generators. Nevertheless there exist many groups, which do not have dead ends of arbitrary depth with respect to any set of generators. Partial results in this direction were obtained by Šunić and by Warshall. We improve these results by showing that abelian groups only ...