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COARSE NON-AMENABILITY AND COVERS WITH SMALL EIGENVALUES GOULNARA ARZHANTSEVA AND ERIK GUENTNER
 

Summary: COARSE NON-AMENABILITY 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 quasi-isometry (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 },

  

Source: Arzhantseva, Goulnara N. - Section de Mathématiques, Université de Genève

 

Collections: Mathematics