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Search: MSC category 20E06 ( Free products, free products with amalgamation, Higman-Neumann-Neumann extensions, and generalizations )

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1. CJM Online first

Graham, Robert; Pichot, Mikael
A Free Product Formula for the Sofic Dimension
It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability measure preserving $s$-regular ergodic discrete groupoids amalgamated over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$ satisfies the equality \[ s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)-\mathfrak{h}(G_3^0)s(G_3) \] where $\mathfrak{h}$ is the normalized Haar measure on $G$.

Keywords:sofic groups, dynamical systems, orbit equivalence, free entropy
Category:20E06

2. CJM 1998 (vol 50 pp. 312)

Dokuchaev, Michael A.; Singer, Maria Lucia Sobral
Units in group rings of free products of prime cyclic groups
Let $G$ be a free product of cyclic groups of prime order. The structure of the unit group ${\cal U}(\Q G)$ of the rational group ring $\Q G$ is given in terms of free products and amalgamated free products of groups. As an application, all finite subgroups of ${\cal U}(\Q G)$, up to conjugacy, are described and the Zassenhaus Conjecture for finite subgroups in $\Z G$ is proved. A strong version of the Tits Alternative for ${\cal U}(\Q G)$ is obtained as a corollary of the structural result.

Keywords:Free Products, Units in group rings, Zassenhaus Conjecture
Categories:20C07, 16S34, 16U60, 20E06

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