A regular map of type $\{m,n\}$ is a 2-cell embedding
of a graph in an orientable surface, with the property
that for any two directed edges $e$ and $e'$ there
exists an orientation-preserving automorphism of the
embedding that takes $e$ onto $e'$, and in which the
face length and the vertex valence are $m$ and $n$,
respectively. Such maps are known to be in a one-to-one
correspondence with torsion-free normal subgroups
of the triangle groups $T(2,m,n)$.
We first show that some of the known existence results
about regular maps follow from residual finiteness of
triangle groups. With the help of representations of
triangle groups in special linear groups over
algebraic extensions of ${\mathbb Z}$ we then
constructively describe homomorphisms from
$T(2,m,n)=\langle y,z|\ y^m=z^n=(yz)^2=1\rangle$
into finite groups of order at most $c^r$ where
$c=c(m,n)$, such that no non-identity word of length
at most $r$ in $x,y$ is mapped onto the identity. As
an application, for any hyperbolic pair $\{m,n\}$ and
any $r$ we construct a finite regular map of type
$\{m,n\}$ of size at most $C^r$, such that every
non-contractible closed curve on the supporting surface
of the map intersects the embedded graph in more than $r$
points. We also show that this result is the best possible
up to determining $C=C(m,n)$. For $r\ge m$ the graphs of
the above regular maps are arc-transitive, of valence $n$,
and of girth $m$; moreover, if each prime divisor of $m$
is larger than $2n$ then these graphs are non-Cayley. 2000 Mathematics Subject Classification:
05C10, 05C25, 20F99, 20H25.