  A split system $\mathcal S$ on a finite set $X$, $|X|\ge3$, is a set of
bipartitions or splits of $X$ which contains all splits of the form
$\{x,X-\{x\}\}$, $x \in X$. To any such split system $\mathcal S$ we can
associate the Buneman graph $\mathcal B(\mathcal S)$ which is essentially a
median graph with leaf-set $X$ that displays the splits in $\mathcal S$. In
this paper, we consider properties of injective split systems, that is, split
systems $\mathcal S$ with the property that $\mathrm{med}_{\mathcal B(\mathcal
S)}(Y) \neq \mathrm{med}_{\mathrm B(\mathcal S)}(Y')$ for any 3-subsets $Y,Y'$
in $X$, where $\mathrm {med}_{\mathcal B(\mathcal S)}(Y)$ denotes the median in
$\mathcal B(\mathcal S)$ of the three elements in $Y$ considered as leaves in
$\mathcal B(\mathcal S)$. In particular, we show that for any set $X$ there
always exists an injective split system on $X$, and we also give a
characterization for when a split system is injective. We also consider how
complex the Buneman graph $\mathcal B(\mathcal S)$ needs to become in order for
a split system $\mathcal S$ on $X$ to be injective. We do this by introducing a
quantity for $|X|$ which we call the injective dimension for $|X|$, as well as
two related quantities, called the injective 2-split and the rooted-injective
dimension. We derive some upper and lower bounds for all three of these
dimensions and also prove that some of these bounds are tight. An underlying
motivation for studying injective split systems is that they can be used to
obtain a natural generalization of symbolic tree maps. An important consequence
of our results is that any three-way symbolic map on $X$ can be represented
using Buneman graphs.
