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Let $G$ be a finitely generated group (optionally torsion-free). Let $N$ be a submonoid of $G$ (that is, a subsemigroup with $1$).

A (cancellative) monoid/semigroup $S$ is right reversible if for all $a,b \in S$, it follows that $Sa \cap Sb \neq\emptyset$. This is the Ore condition for semigroups, and allows one to invert the elements of $S$ so that the set of products, $S\cdot S^{-1}$, is a group.

Suppose that $G$ is nilpotent. Does it follow that every submonoid $N$ is right reversible?

What are the right conditions on $G$ to guarantee this property?

Something like nilpotent by finite comes to mind, since this implies the group ring ${\bf Z} G$ is Goldie. Obviously, this is a strong condition on $G$, since it fails for large classes of groups, e.g., those containing a non-abelian free group, or probably, those for which the group ring is not Goldie.

The question arose from a study of random walks on discrete groups, related to the boundary of one of the usual boundaries. It is a little too complicated to explain the connections here.

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It follows from Lemma 2 in Grigorchuk's https://page-one.live.cf.public.springer.com/pdf/preview/10.1007/BF01138837 that all subsemigroups of a group $G$ are right reversible iff it contains no free semigroup on 2 generators.

A free subsemigroup is not right reversible. Conversely suppose $S$ is not right reversible (say $aS\cap bS=\emptyset$, note for me right reversible is dual to the OP's definition). Then observe that $a,b$ generate a free subsemigroup. For words beginning with distinct letters are not equal and if they begin with the same letter you can cancel it to get a shorter relation.

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    $\begingroup$ And from Theorem 1, necessary and sufficient is that the group be almost nilpotent. Thanks, Ben. $\endgroup$ – David Handelman Jul 29 '17 at 18:47
  • $\begingroup$ Yes. Any group of subexponential growth has this property. $\endgroup$ – Benjamin Steinberg Jul 29 '17 at 18:53
  • $\begingroup$ My colleague, Vadim Kaimanovich, just told me that the result also follows from an argument in his 1985 thesis. $\endgroup$ – David Handelman Jul 29 '17 at 23:09
  • $\begingroup$ Could be. I think Grigorchuk also announced this in a 1985 conference proceedings $\endgroup$ – Benjamin Steinberg Jul 29 '17 at 23:34

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