While finite type Artin groups and right-angled Artin groups are well-understood, little is known about more general Artin groups. In this paper we use the action of an infinite type Artin group AΓ on a CAT(0) cube complex to prove that AΓ has trivial center providing Γ is not the star of a single vertex, and is acylindrically hyperbolic providing Γ is not a join.

No abstract

We investigate algorithmic control of a large swarm of mobile particles (such as robots, sensors, or building material) that move in a 2D workspace using a global input signal (such as gravity or a magnetic field). Upon activation of the field, each particle moves maximally in the same direction until forward progress is blocked by a stationary obstacle or another stationary particle. In an open workspace, this system model is of limited use because it has only two controllable degrees of freedom-all particles receive the same inputs and move uniformly. We show that adding a maze of obstacles to the environment can make the system drastically more complex but also more useful.We provide a wide range of results for a wide range of questions. These can be subdivided into external algorithmic problems, in which particle configurations serve as input for computations that are performed elsewhere, and internal logic problems, in which the particle configurations themselves are used for carrying out computations.For external algorithms, we give both negative and positive results. If we are given a set of stationary obstacles, we prove that it is NP-hard to decide whether a given initial configuration of unit-sized particles can be transformed into a desired target configuration. Moreover, we show that finding a control sequence of minimum length is PSPACE-complete. We also work on the inverse problem, providing constructive algorithms to design workspaces that efficiently implement arbitrary permutations between different configurations.For internal logic, we investigate how arbitrary computations can be implemented. We demonstrate how to encode dual-rail logic to build a universal logic gate that concurrently evaluates and, nand, nor, and or operations. Using many of these gates and appropriate interconnects, we can evaluate any logical expression. However, we establish that simulating the full range of complex interactions present in arbitrary digital circuits encounters a fundamental difficulty: a fan-out gate cannot be generated. We resolve this missing component with the help of 2×1 particles, which can create fan-out gates that produce multiple copies of the inputs. Using these gates we provide rules for replicating arbitrary digital circuits.

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed using parabolic subgroups. We extend the construction of the complex of parabolic subgroups to FC type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is δ−hyperbolic.

We present fundamental progress on the computational universality of swarms of micro-or nano-scale robots in complex environments, controlled not by individual navigation, but by a uniform global, external force. Consider a 2D grid world, in which all obstacles and robots are unit squares, and for each actuation, robots move maximally until they collide with an obstacle or another robot. In previous work, we demonstrated components of particle computation in this world, designing obstacle configurations that implement AND and OR logic gates: by using dual-rail logic, we designed NOT, NOR, NAND, XOR, XNOR logic gates. However, we were unable to design a FAN-OUT gate, which is necessary for simulating the full range of complex interactions that are present in arbitrary digital circuits. In this work we resolve this problem by proving unit-sized robots cannot generate a FAN-OUT gate. On the positive side, we resolve the missing component with the help of 2×1 robots, which can create fan-out gates that produce multiple copies of the inputs. Using these gates we are able to establish the full range of computational universality as presented by complex digital circuits. As an example we connect our logic elements to produce a 3-bit counter. We also demonstrate how to implement a data storage element.