On the rank of intersection of subgroups of a free product of groups

On the Kurosh rank of the intersection of subgroups in free products of groups

“…This example proves that the main result (5) of article [3] is incorrect. The aim of this article is to make another attempt to improve on Soma's bound (4) and to generalize it for a free product * α∈I G α of arbitrary family of groups G α , α ∈ I .…”

“…Note that our version, cf. [3,11,12,19], enables us to establish the estimate (9) for a free product * α∈I G α of an arbitrary family of groups G α , α ∈ I , whereas results of [3,19] apply only to a free product A * B of two groups A, B. In particular, Burns, Chau and Kam [3] conjectured that some version of their estimate (5) would also hold for a free product of not necessarily two factors and our results fully confirm this conjecture.…”

“…However, B. Baumslag did not give any estimate of type (1). A version of the estimate (1) for free products was obtained by Soma [19] in 1990 and an attempt to improve on the Soma's estimate was made by Burns, Chau and Kam [3] in 1998.…”

“…To state results of [3,19], we recall that, according to the classic Kurosh subgroup theorem [13] proved in 1934, see also [14,15], every subgroup H of a free product * α∈I G α of groups G α , α ∈ I , itself is a free product…”

“…In 1998, Burns, Chau and Kam [3] introduced the term of the Kurosh rank, observed that Soma's results [19] imply the inequality (4) and attempted to improve on (4) and to show that…”