Approximate centralizer

Definition with symbols
Let $$G$$ be a group and $$g \in G$$ be an element. Then, the approximate centralizer of g in $$G$$, denoted as $$C_G^*(g)$$, is defined as the union of centralizers of all positive (or equivalently, all nonzero) powers of $$g$$.

The approximate centralizer is always a subgroup, and it contains the centralizer. When $$g$$ has finite order, its approximate centralizer is the whole group. Hence, the approximate centralizer makes sense, or is worth studying, only for elements of infinite order.