| Summary: | The group <i>X</i>(<i>G</i>) is obtained from <i>G</i>∗<i>G</i> by forcing each element <i>g</i> in the first free factor to commute with the copy of <i>g</i> in the second free factor. We make significant additions to the list of properties that the functor <i>X</i> is known to preserve. We also investigate the geometry and complexity of the word problem for <i>X</i>(<i>G</i>). Subtle features of <i>X</i>(<i>G</i>) are encoded in a normal abelian subgroup <i>W</i><<i>X</i>(<i>G</i>) that is a module over <i>ZQ</i>, where <i>Q</i>=<i>H</i><sub>1</sub> (<i>G</i>,<i>Z</i>). We establish a structural result for this module and illustrate its utility by proving that <i>X</i> preserves virtual nilpotence, the Engel condition, and growth type – polynomial, exponential, or intermediate. We also use it to establish isoperimetric inequalities for <i>X</i>(<i>G</i>) when <i>G</i> lies in a class that includes Thompson's group <i>F</i> and all non-fibred Kähler groups. The word problem is soluble in <i>X</i>(<i>G</i>) if and only if it is soluble in <i>G</i>. The Dehn function of <i>X</i>(<i>G</i>) is bounded below by a cubic polynomial if <i>G</i> maps onto a non-abelian free group.
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