On structures for free-by-cyclic groups

We construct CAT ( 0 ) spaces on which various free-by-cyclic groups act. Let G be the free-by-cyclic group F ( a , c 1 , … , c n , d ) ⋊ φ Z with φ determined by ϕ ( a ) = a , ϕ ( c j ) = c j a κ , and ϕ ( d ) = d w , where w is some word not containing d. Our main result is that if the exponent sum of c n in w is non-zero, then there is a CAT ( 0 ) metric space on which G acts properly discontinuously and cocompactly by isometries.