Residual Properties of Free Products of Infinitely Many Nilpotent Groups Amalgamating Cycles

Abstract

We prove that a free product of infinitely many finitely generated, torsion free, nilpotent groups amalgamating isolated cycles is residually finite-p for all primes p so long as the number of generators and nilpotency class of the factors are both bounded. Examples illustrate that if we remove any single hypothesis, the resulting group need not even be residually finite.

Residual P-Finiteness of Certain Generalized Free Products of Nilpotent Groups

“…= y 2 and y 1 = x 2 2 is not RF 2[5]. Moreover A 1 , A 2 are finite 2-groups of order 8 and H = x2 1 , y 1 is order 4. Thus H is normal in both A 1 and A 2 .…”

“…Thus H is normal in both A 1 and A 2 . Clearly 1 = x2 1 ∈ H. Note that H has nontrivial subgroups x2 1 , y 1 , x 2 1 y 1 . But y 1 , x 2 1 y 1 are not normal in A 1 .…”