67,339 research outputs found
Knapsack Problems in Groups
We generalize the classical knapsack and subset sum problems to arbitrary
groups and study the computational complexity of these new problems. We show
that these problems, as well as the bounded submonoid membership problem, are
P-time decidable in hyperbolic groups and give various examples of finitely
presented groups where the subset sum problem is NP-complete.Comment: 28 pages, 12 figure
Stable commutator length in Baumslag-Solitar groups and quasimorphisms for tree actions
This paper has two parts, on Baumslag-Solitar groups and on general G-trees.
In the first part we establish bounds for stable commutator length (scl) in
Baumslag-Solitar groups. For a certain class of elements, we further show that
scl is computable and takes rational values. We also determine exactly which of
these elements admit extremal surfaces.
In the second part we establish a universal lower bound of 1/12 for scl of
suitable elements of any group acting on a tree. This is achieved by
constructing efficient quasimorphisms. Calculations in the group BS(2,3) show
that this is the best possible universal bound, thus answering a question of
Calegari and Fujiwara. We also establish scl bounds for acylindrical tree
actions.
Returning to Baumslag-Solitar groups, we show that their scl spectra have a
uniform gap: no element has scl in the interval (0, 1/12).Comment: v2: minor changes, incorporates referee suggestions; v1: 36 pages, 10
figure
On the automorphism group of generalized Baumslag-Solitar groups
A generalized Baumslag-Solitar group (GBS group) is a finitely generated
group which acts on a tree with all edge and vertex stabilizers infinite
cyclic. We show that Out(G) either contains non-abelian free groups or is
virtually nilpotent of class at most 2. It has torsion only at finitely many
primes.
One may decide algorithmically whether Out(G) is virtually nilpotent or not.
If it is, one may decide whether it is virtually abelian, or finitely
generated. The isomorphism problem is solvable among GBS groups with Out(G)
virtually nilpotent.
If is unimodular (virtually ), then Out(G) is commensurable
with a semi-direct product with virtually free
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