# Virtually abelian centralizers

This is a sort of a follow-up question to Limits of conjugated subgroups (though it might not seem at first glance to have much to do with it.)

Anyway, I'm wondering what sort of groups have the property that the centralizer of an element of infinite order is virtually abelian. For example hyperbolic groups are known to have this property: the infinite cyclic subgroup $\langle g\rangle$ has finite index in $C(g)$.

That's about the extent of my knowledge in this question.

In particular I'm interested in groups acting by isometries on a CAT(0) space (especially fundamental groups of NPC-spaces), semihyperbolic groups etc.

Of course, any information someone might be able to provide is greatly appreciated.

• All subgroups of $SL_2(\mathbf{C})$ have this property. – YCor Feb 9 '15 at 18:43
• Or ${\mathbb Z} \wr {\mathbb Z}$ (the restricted wreath product). – Derek Holt Feb 9 '15 at 18:49
• Commutative transitive groups (i.e. those where all centralizers of all non-identity elements are abelian) would in particular have this property: this may not give new examples that have not already been mentioned, but may prove useful for looking things up in the literature – Yemon Choi Feb 9 '15 at 19:07
• @YemonChoi Some Olshanski monsters are commutative transitive and certainly not subgroups of $SL_2$. Also using that commutative transitive is stable under free product provides further examples. – YCor Feb 9 '15 at 21:01
• Thompson's group $F$ acts properly on a (complete) CAT(0) cube complex, but the centralizer of an element of $F$ is the product of a free abelian group with some isomorphic copies of $F$ (according to Guba and Sapir). – Seirios Feb 10 '15 at 18:23