  Relativizing computations of Turing machines to an oracle is a central
concept in the theory of computation, both in complexity theory and in
computability theory(!). Inspired by lowness notions from computability theory,
Allender introduced the concept of "low for speed" oracles. An oracle A is low
for speed if relativizing to A has essentially no effect on computational
complexity, meaning that if a decidable language can be decided in time $f(n)$
with access to oracle A, then it can be decided in time poly(f(n)) without any
oracle. The existence of non-computable such A's was later proven by Bayer and
Slaman, who even constructed a computably enumerable one, and exhibited a
number of properties of these oracles as well as interesting connections with
computability theory. In this paper, we pursue this line of research, answering
the questions left by Bayer and Slaman and give further evidence that the
structure of the class of low for speed oracles is a very rich one.
