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A Coxeter system is an ordered pair (W, S) where S is the generating set in a particular type of presentation for the Coxeter group W . A subgroup of W is called special if it is generated by a subset of S. Amalgamated product decompositions of a Coxeter group having special factors and special amalgamated subgroup are easily recognized from the presentation of the Coxeter group. If a Coxeter group is a subgroup of the fundamental group of a given graph of groups, then the Coxeter group is also the fundamental group of a graph of special subgroups, where each vertex and edge group is a subgroup of a conjugate of a vertex or edge group of the given graph of groups. A vertex group of an arbitrary graph of groups decomposition of a Coxeter group is shown to split into parts conjugate to special groups and parts that are subgroups of edge groups of the given decomposition. Several applications of the main theorem are produced, including the classification of maximal FA-subgroups of a finitely generated Coxeter group as all conjugates of certain special subgroups.

We study the relationship between two sets S and S 0 of Coxeter generators of a finitely generated Coxeter group W by proving a series of theorems that identify common features of S and S 0 . We describe an algorithm for constructing from any set of Coxeter generators S of W a set of Coxeter generators R of maximum rank for W .A subset C of S is called complete if any two elements of C generate a finite group. We prove that if S and S 0 have maximum rank, then there is a bijection between the complete subsets of S and the complete subsets of S 0 so that corresponding subsets generate isomorphic Coxeter systems. In particular, the Coxeter matrices of .W; S/ and .W; S 0 / have the same multiset of entries.

This paper analyzes the effects of mergers between firms competing by simultaneously choosing price and location. Products combined by a merger are repositioned away from each other to reduce cannibalization, and non-merging substitutes are, in response, repositioned between the merged products. This repositioning greatly reduces the merged firm's incentive to raise prices and thus substantially mitigates the anticompetitive effects of the merger. Computation of, and selection among, equilibria is done with a novel technique known as the stochastic response dynamic, which does not require the computation of first-order conditions.