A surjunctive group is a group 
for which every injective cellular automaton with a finite
set of states is surjective (Ceccherini-Silberstein
and Coornaert 2010). More precisely, for every nonempty finite set 
, consider the space 
of configurations, meaning all functions 
. Give 
the discrete topology
and 
the product topology. The left shift group
action is
 |
The group 
is surjunctive when every injective,
continuous equivariant
map 
is surjective. Here equivariance
means 
for every 
. These continuous equivariant maps are the cellular
automata indexed by 
.
Every finite group is surjunctive because 
is a finite set and an injective self-map of a finite
set is surjective. Every sofic
group is also surjunctive (Gromov 1999, Weiss 2000). The assertion that every
group is surjunctive is the Gottschalk
surjunctivity conjecture (Gottschalk 1973).
