 
Summary: Zariski dense surface subgroups in SL(3, Z).
D. D. LONG
A.W. REID
M. THISTLETHWAITE
We exhibit an infinite family of Zariskidense surface groups of fixed genus inside
SL(3, Z).
22E40, 20H10
1 Introduction
This paper proves the existence of an infinite family of Zariskidense surface subgroups
of fixed genus inside SL(3, Z). To put this into context, we recall that it follows from
Tits [11] that free groups are plentiful in SL(3, Z) and, moreover, it is not difficult to
see that (one can use Theorem 2.2 below, for example) it is easily arranged that these
free groups are Zariski dense. Less trivially, classical arithmetic considerations (see for
example §6.1 of [7]) can be used to construct surface subgroups of SL(3, Z) of every
genus 2. However these are constructed using the theory of quadratic forms, so that
their Zariski closures in SL(3, R) are SO(f, R) for some appropriate ternary quadratic
form f ; in particular these surface groups are not Zariski dense in SL(3, R). With this
as background we state the main result of this note:
Theorem 1.1 The family of representations of the triangle group
(3, 3, 4) =< a, b  a3
