On free products with amalgamation of two infinite cyclic groups

Discrete Free Products of Two Complex Cyclic Matrix Groups

Observe that the elements of ftxUf^ have infinite order, while those of ft2Uft3 have finite order. We will prove that whenever A e ft and B e ft do not both have a fixed point on L, then the group (A,B') is the discrete free product (A)*(B'), where B' denotes the transpose of B. The case where both A and B have a fixed point on L is also discussed. We show in addition that if A eft and Beft are...

On the Cyclic Subgroup Separability of Free Products of Two Groups with Amalgamated Subgroup

Let G be a free product of two groups with amalgamated subgroup, π be either the set of all prime numbers or the one-element set {p} for some prime number p. Denote by Σ the family of all cyclic subgroups of group G, which are separable in the class of all finite π-groups. Obviously, cyclic subgroups of the free factors, which aren’t separable in these factors by the family of all normal subgro...

Free Products with Amalgamation and 3-manifolds

Random walks on free products of cyclic groups

Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbor random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, th...

Ultrafilters and Partial Products of Infinite Cyclic Groups

We consider, for infinite cardinals κ and α ≤ κ+, the group Π(κ,<α) of sequences of integers, of length κ, with non-zero entries in fewer than α positions. Our main result tells when Π(κ,<α) can be embedded in Π(λ,<β). The proof involves some set-theoretic results, one about families of finite sets and one about families of ultrafilters. Partially supported by NSF grant DMS–0070723. Part of thi...