 
Summary: COARSE NONAMENABILITY AND COVERS WITH SMALL EIGENVALUES
GOULNARA ARZHANTSEVA AND ERIK GUENTNER
ABSTRACT. Given a closed Riemannian manifold M and a (virtual) epimorphism 1(M) F2
of the fundamental group onto a free group of rank 2, we construct a tower of finite sheeted regular
covers {Mn}
n=0 of M such that 1(Mn) 0 as n . This is the first example of such a tower
which is not obtainable up to uniform quasiisometry (or even up to uniform coarse equivalence) by
the previously known methods where 1(M) is supposed to surject onto an amenable group.
1. INTRODUCTION
Let M be a closed (that is, compact and without boundary) Riemannian manifold with funda
mental group 1(M). A residually finite group G, a surjective homomorphism 1(M) G and a
nested sequence of finite index normal subgroups of G with trivial intersection gives rise to a tower
of finite sheeted regular covers of M; conversely, every tower of finite sheeted regular covers arises
in this manner. In summary, writing G0 = G and M0 = M, we have:
()
G0 G1 G2 · · · , with
n=0
Gn = { 1 },
