  The study of the word problems of groups dates back to Dehn in 1911, and has
been a central topic of study in both group theory and computability theory. As
most naturally occurring presentations of groups are recursive, their word
problems can be thought of as a computably enumerable equivalence relation
(ceer). In this paper, we study the word problem of groups in the framework of
ceer degrees, introducing a new metric with which to study word problems. This
metric is more refined than the classical context of Turing degrees.
  Classically, every Turing degree is realized as the word problem of some c.e.
group, but this is not true for ceer degrees. This motivates us to look at the
classical constructions and show that there is a group whose word problem is
not universal, but becomes universal after taking any nontrivial free product,
which we call $*$-universal. This shows that existing constructions of the
Higman embedding theorem do not preserve ceer degrees. We also study the index
set of various classes of groups defined by their properties as a ceer: groups
whose word problems are dark (equivalently, algorithmically finite as defined
by Miasnikov and Osin), universal, and $*$-universal groups.
