Geodesic
Large hyperbolic lattices
Algebra i logika, Tome 48 (2009) no. 2, pp. 174-189

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For a fundamental group of a compact orientable manifold, a condition is specified that is sufficient to guarantee the presence of a “virtual” epimorphism onto a free non-Abelian group. A consequence is deriving a strong Tits alternative. An arbitrary noncompact finitely generated discrete subgroup in $\mathrm{PO}(3,1)$ either is large or is virtually Abelian. An application is provided to the problem of uniform exponential growth for lattices in a 3-dimensional hyperbolic space and of growth of Betti numbers for lattices in a hyperbolic $n$-dimensional space, where $n$ is an odd number.
Keywords: fundamental group, compact orientable manifold, discrete subgroup, hyperbolic lattice, uniform exponential growth problem. 
@article{AL_2009_48_2_a1,
     author = {F. Grunewald and G. A. Noskov},
     title = {Large hyperbolic lattices},
     journal = {Algebra i logika},
     pages = {174--189},
     publisher = {mathdoc},
     volume = {48},
     number = {2},
     year = {2009},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/AL_2009_48_2_a1/}
}
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F. Grunewald; G. A. Noskov. Large hyperbolic lattices. Algebra i logika, Tome 48 (2009) no. 2, pp. 174-189. http://geodesic.mathdoc.fr/item/AL_2009_48_2_a1/