283 research outputs found

    Rational subsets of Baumslag-Solitar groups

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    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 GL(2,Q)\text{GL}(2,\mathbb{Q}). We show that rational subset membership for Baumslag-Solitar groups BS(1,q)\text{BS}(1,q) with q≥2q\ge 2 is decidable and PSPACE-complete. To this end, we introduce a word representation of the elements of BS(1,q)\text{BS}(1,q): their pointed expansion (PE), an annotated qq-ary expansion. Seeing subsets of BS(1,q)\text{BS}(1,q) as word languages, this leads to a natural notion of PE-regular subsets of BS(1,q)\text{BS}(1, q): these are the subsets of BS(1,q)\text{BS}(1,q) whose sets of PE are regular languages. Our proof shows that every rational subset of BS(1,q)\text{BS}(1,q) is PE-regular. Since the class of PE-regular subsets of BS(1,q)\text{BS}(1,q) 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 BS(1,q)\text{BS}(1,q) 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 BS(1,q)\text{BS}(1,q) has finite index.Comment: Long version of paper with same title appearing in ICALP'2

    Tiling Problems on Baumslag-Solitar groups

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    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

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    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

    The large scale geometry of some metabelian groups

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    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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