283 research outputs found
Rational subsets of Baumslag-Solitar groups
We consider the rational subset membership problem for Baumslag-Solitar
groups. These groups form a prominent class in the area of algorithmic group
theory, and they were recently identified as an obstacle for understanding the
rational subsets of .
We show that rational subset membership for Baumslag-Solitar groups
with is decidable and PSPACE-complete. To this end,
we introduce a word representation of the elements of : their
pointed expansion (PE), an annotated -ary expansion. Seeing subsets of
as word languages, this leads to a natural notion of
PE-regular subsets of : these are the subsets of
whose sets of PE are regular languages. Our proof shows that
every rational subset of is PE-regular.
Since the class of PE-regular subsets of is well-equipped
with closure properties, we obtain further applications of these results. Our
results imply that (i) emptiness of Boolean combinations of rational subsets is
decidable, (ii) membership to each fixed rational subset of is
decidable in logarithmic space, and (iii) it is decidable whether a given
rational subset is recognizable. In particular, it is decidable whether a given
finitely generated subgroup of has finite index.Comment: Long version of paper with same title appearing in ICALP'2
Tiling Problems on Baumslag-Solitar groups
We exhibit a weakly aperiodic tile set for Baumslag-Solitar groups, and prove
that the domino problem is undecidable on these groups. A consequence of our
construction is the existence of an arecursive tile set on Baumslag-Solitar
groups.Comment: In Proceedings MCU 2013, arXiv:1309.104
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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The large scale geometry of some metabelian groups
We study the large scale geometry of the upper triangular subgroup of
PSL(2,Z[1/n]), which arises naturally in a geometric context. We prove a
quasi-isometry classification theorem and show that these groups are
quasi-isometrically rigid with infinite dimensional quasi-isometry group. We
generalize our results to a larger class of groups which are metabelian and are
higher dimensional analogues of the solvable Baumslag-Solitar groups BS(1,n)
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