Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups

We prove by explicit construction that graph braid groups and most surface groups can be embedded in a natural way in right-angled Artin groups, and we point out some consequences of these embedding results. We also show that every right-angled Artin group can be embedded in a pure surface braid group. On the other hand, by generalising to right-angled Artin groups a result of Lyndon for free groups, we show that the Euler characteristic -1 surface group (given by the relation x^2y^2=z^2) never embeds in a right-angled Artin group.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-22.abs.htm

Embedding right-angled Artin groups into graph braid groups

We construct an embedding of any right-angled Artin group $G(\Delta)$ defined by a graph $\Delta$ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of $\Delta$. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.Comment: 8 pages. Final version, appears in Geometriae Dedicata

Parabolic isometries of CAT(0) spaces and CAT(0) dimensions

We study discrete groups from the view point of a dimension gap in connection to CAT(0) geometry. Developing studies by Brady-Crisp and Bridson, we show that there exist finitely presented groups of geometric dimension 2 which do not act properly on any proper CAT(0) spaces of dimension 2 by isometries, although such actions exist on CAT(0) spaces of dimension 3. Another example is the fundamental group, G, of a complete, non-compact, complex hyperbolic manifold M with finite volume, of complex-dimension n > 1. The group G is acting on the universal cover of M, which is isometric to H^n_C. It is a CAT(-1) space of dimension 2n. The geometric dimension of G is 2n-1. We show that G does not act on any proper CAT(0) space of dimension 2n-1 properly by isometries. We also discuss the fundamental groups of a torus bundle over a circle, and solvable Baumslag-Solitar groups.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol4/agt-4-38.abs.htm

The conjugacy problem in right-angled Artin groups and their subgroups

29 pages, 7 figuresInternational audienceWe prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes'' studied independently by Haglund and Wise

Morse theory and conjugacy classes of finite subgroups II

We construct a hyperbolic group with a finitely presented subgroup, which has infinitely many conjugacy classes of finite-order elements. We also use a version of Morse theory with high dimensional horizontal cells and use handle cancellation arguments to produce other examples of subgroups of CAT(0) groups with infinitely many conjugacy classes of finite-order elements.Comment: 18 pages, 7 figure