Description:
<jats:title>Abstract</jats:title><jats:p>A word <jats:italic>W</jats:italic> in a group <jats:italic>G</jats:italic> is a geodesic (weighted geodesic) if <jats:italic>W</jats:italic> has minimum length (minimum weight with respect to a generator weight function α) among all words equal to <jats:italic>W</jats:italic>. For finitely generated groups, the word problem is equivalent to the geodesic problem. We prove: (i) There exists a group <jats:italic>G</jats:italic> with solvable word problem, but unsolvable geodesic problem, (ii) There exists a group <jats:italic>G</jats:italic> with a solvable weighted geodesic problem with respect to one weight function α<jats:sub>1</jats:sub>, but unsolvable with respect to a second weight function α<jats:sub>2</jats:sub>. (iii) The (ordinary) geodesic problem and the free-product geodesic problem are independent.</jats:p>