The homology cylinder group is the monoid of homology cylinders modulo homology cobordism. I wonder whether there is any known finite order element in this group.



Is there a torsion element in the homology cylinder group?
Certainly there is a lot of $2$-torsion. One easy way to construct such an example is to look at the embedding of the string link cobordism group on $n$ strands into the homology cobordism group of genus $n$. (Levine explains this embedding in his paper "Homology cylinders: an enlargment of the mapping class group." See Theorem 4 of that paper.) The string link group has lots of $2$-torsion. For example just tie an amphicheiral knot (which is not slice) into one of the strands. In fact the knot concordance group has infinitely many $\mathbb Z\_2$ summands which inject into the homology cylinder group.


