464 research outputs found
Virtually splitting the map from Aut(G) to Out(G)
We give an elementary criterion on a group G for the map from Aut(G) to
Out(G) to split virtually. This criterion applies to many residually finite
CAT(0) groups and hyperbolic groups, and in particular to all finitely
generated Coxeter groups. As a consequence the outer automorphism group of any
finitely generated Coxeter group is residually finite and virtually
torsion-free.Comment: 10 pages, 1 figur
The Word Problem in the Baumslag group with a non-elementary Dehn function is polynomial time decidable
We prove that the Word problem in the Baumslag group G(1,2) which has a
non-elementary Dehn function is decidable in polynomial time
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
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Decision problems and profinite completions of groups
We consider pairs of finitely presented, residually finite groups
P\hookrightarrow\G for which the induced map of profinite completions \hat
P\to \hat\G is an isomorphism. We prove that there is no algorithm that, given
an arbitrary such pair, can determine whether or not is isomorphic to \G.
We construct pairs for which the conjugacy problem in \G can be solved in
quadratic time but the conjugacy problem in is unsolvable.
Let be the class of super-perfect groups that have a compact
classifying space and no proper subgroups of finite index. We prove that there
does not exist an algorithm that, given a finite presentation of a group \G
and a guarantee that \G\in\mathcal J, can determine whether or not
\G\cong\{1\}.
We construct a finitely presented acyclic group \H and an integer such
that there is no algorithm that can determine which -generator subgroups of
\H are perfect
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