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We introduce the notion of halfspaces associated to a group splitting, andinvestigate the relationship between the coarse geometry of the halfspaces andthe coarse geometry of the group. Roughly speaking, the halfspaces of a groupsplitting are subgraphs of the Cayley graph obtained by pulling back thehalfspaces of the Bass--Serre tree. Our first theorem shows that (under mildconditions) any splitting of a one-ended group can be upgraded to a splittingwhere all the halfspaces are one-ended. Our second theorem demonstrates that aone-ended group usually has a JSJ splitting where all the halfspaces areone-ended. And our third theorem states that if a one-ended finitely presentedgroup $G$ admits a splitting such that some edge stabilizer has more than oneend, but the halfspaces associated to the edge stabilizer are one-ended, then$H^2(G,\mathbb ZG)\ne \{0\}$; in particular $G$ is not simply connected atinfinity and $G$ is not an $n$-dimensional duality group for $n\geq3$.

Splittings of One-Ended Groups with One-Ended Halfspaces