252 research outputs found
New lower bounds on subgroup growth and homology growth
We establish new strong lower bounds on the (subnormal) subgroup growth of a
large class of groups. This includes the fundamental groups of all
finite-volume hyperbolic 3-manifolds and all (free non-abelian)-by-cyclic
groups. The lower bound is nearly exponential, which should be compared with
the fastest possible subgroup growth of any finitely generated group. This is
achieved by free non-abelian groups and is slightly faster than exponential. As
a consequence, we obtain good estimates on the number of covering spaces of a
hyperbolic 3-manifold with given covering degree. We also obtain slightly
weaker information on the number of covering spaces of closed 4-manifolds with
non-positive Euler characteristic. The results on subgroup growth follow from a
new theorem which places lower bounds on the rank of the first homology (with
mod p coefficients) of certain subgroups of a group. This is proved using a
topological argument.Comment: 39 pages, 2 figures; v3 has minor changes from v2, incorporating
referee's comments; v2 has minor changes from v1; to appear in the
Proceedings of the London Mathematical Societ
Two mod-p Johnson filtrations
We consider two mod-p central series of the free group given by Stallings and
Zassenhaus. Applying these series to definitions of Dennis Johnson's filtration
of the mapping class group we obtain two mod-p Johnson filtrations. Further, we
adapt the definition of the Johnson homomorphisms to obtain mod-p Johnson
homomorphisms.
We calculate the image of the first of these homomorphisms. We give
generators for the kernels of these homomorphisms as well. We restrict the
range of our mod-p Johnson homomorphisms using work of Morita. We finally prove
the announced result of Perron that a rational homology 3-sphere may be given
as a Heegaard splitting with gluing map coming from certain members of our
mod-p Johnson filtrations.Comment: 35 pages, 1 figure; added referenc
Cutting up graphs revisited - a short proof of Stallings' structure theorem
This is a new and short proof of the main theorem of classical structure tree
theory. Namely, we show the existence of certain automorphism-invariant
tree-decompositions of graphs based on the principle of removing finitely many
edges. This was first done in "Cutting up graphs" by M.J. Dunwoody. The main
ideas are based on the paper "Vertex cuts" by M.J. Dunwoody and the author. We
extend the theorem to a detailed combinatorial proof of J.R. Stallings' theorem
on the structure of finitely generated groups with more than one end.Comment: 12 page
Hyperbolic groups that are not commensurably coHopfian
Sela proved every torsion-free one-ended hyperbolic group is coHopfian. We
prove that there exist torsion-free one-ended hyperbolic groups that are not
commensurably coHopfian. In particular, we show that the fundamental group of
every simple surface amalgam is not commensurably coHopfian.Comment: v3: 14 pages, 4 figures; minor changes. To appear in International
Mathematics Research Notice
Subgroups of direct products of elementarily free groups
We exploit Zlil Sela's description of the structure of groups having the same
elementary theory as free groups: they and their finitely generated subgroups
form a prescribed subclass E of the hyperbolic limit groups.
We prove that if are in E then a subgroup is of type \FP_n if and only if is itself,
up to finite index, the direct product of at most groups from .
This answers a question of Sela.Comment: 19 pages, no figure
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