11,838 research outputs found
On the cardinality of sumsets in torsion-free groups
Let be finite subsets of a torsion-free group . We prove that for
every positive integer there is a such that if then
the inequality holds unless a left translate of is
contained in a cyclic subgroup. We obtain for arbitrary
torsion-free groups, and for groups with the unique product
property, where is an absolute constant. We give examples to show that
is at least quadratic in
Finite index subgroups without unique product in graphical small cancellation groups
We construct torsion-free hyperbolic groups without unique product whose
subgroups up to some given finite index are themselves non-unique product
groups. This is achieved by generalising a construction of Comerford to
graphical small cancellation presentations, showing that for every subgroup
of a graphical small cancellation group there exists a free group such that
admits a graphical small cancellation presentation.Comment: 8 pages, 1 figur
Approximate groups, I: the torsion-free nilpotent case
We describe the structure of ``K-approximate subgroups'' of torsion-free
nilpotent groups, paying particular attention to Lie groups.
Three other works, by Fisher-Katz-Peng, Sanders and Tao, have appeared which
independently address related issues. We comment briefly on some of the
connections between these papers.Comment: 23 page
On residualizing homomorphisms preserving quasiconvexity
H is called a G-subgroup of a hyperbolic group G if for any finite subset M G there exists a homomorphism from G onto a non-elementary hyperbolic group G_1 that is surjective on H and injective on M. In his paper in 1993 A. Ol'shanskii gave a description of all G-subgroups in any given non-elementary hyperbolic group G. Here we show that for the same class of G-subgroups the finiteness assumption on M (under certain natural conditions) can be replaced by an assumption of quasiconvexity
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