Hyperbolic groups with $k$-geodetic Cayley graphsĪ locally-finite simple connected graph is said to be $k$-geodetic for some $k\geq1$, if there is at most $k$ distinct geodesics between any two vertices of the graph. This work extends results of Holt and Plesken from 1989 and illustrates the scope of algorithmic improvements over the past decades. It crucially relies on new tools for calculating cohomology, as well as improved implementations for isomorphism test. I will describe a recent project to enumerate, up to isomorphism, the perfect groups of order up to $2\cdot 10^6$. The construction of perfect groups of a given order can be considered as the prototype of the construction of nonsolvable groups of a given order. Recordings of talks are collected on YouTube. Getting there: The venue (search for room) on the Callaghan campus of The University of Newcastle can be reached in at least three ways: By bus, going to " Mathematics Building, Ring Rd" by train, going to " Warabrook Station" and walking about 15-20 minutes across the campus or by car and parking, e.g., in carpark " P2". To be added to or removed from the mailing list, or for any other information, please contact Michal Ferov. The mailing list carries updates and reminders as new seminars are announced. It's also a great opportunity to discover why Newcastle is Australia's coolest coastal town! Topics of interest include all aspects of group theory and connections to computer science, dynamics, graph theory, logic, number theory, operator algebras, topology.Īll interested are warmly invited to attend.
SUBSHIFT SRT SERIES
This is typified by the Morse shift.Įxcept for trivial cases, SFT's and sofic shifts are not minimal.This is a series of meetings with the aim of bringing together mathematicians working on Symmetry - broadly understood - that are based around Newcastle - also broadly understood. Of particular interest are minimal shift spaces, those in which every point hasĪ dense forward orbit. However, these classes barely scratch the surface of the range of behaviours exhibited by SFT's and sofic shifts are useful in modeling dynamical systems and applications to data recording. These results areĬlosely connected to results in automata theory (Béal ,
Unique minimal right resolving presentation. Presented in such a way, and every irreducible sofic shift has a Outgoing edges have distinct labels (as inFigure 4). It is often easier to work with presentations whichĪre right resolving, meaning that at any given state, all Is not hard to see that the Morse shift is not sofic.Ī labelled graph that presents a sofic shift is called a presentation. Prime shift is not sofic (Aho, Hopcroft, and Ullman ). Version of the pumping lemma from automata theory shows that the SFT'sĪre special kinds of sofic shifts any \(M\)-step SFT can be presentedīy an edge-labelled graph whose states are the allowed \(M\)-blocks. Labelled graph, vertices can be viewed as state information whichĬonnects sequences in the past with sequences in the future. Sofic shifts can be regarded as "finite-state systems": in a Vertices) are allowed to have the same label. The even shift is not an SFT, it is a sofic shift, as presented inįigure 4. Vertex) labelled by a symbol from an alphabet, the set ofīi-infinite label sequences of paths in \(G\) is a sofic shift, andĮvery sofic shift is conjugate to one presented in this way. Equivalently, given a graph \(G\) with each edge (or Subshifts (called irreducible components).Ī sofic shift is a shift space that is a factor of an SFT One can study a general SFT by studying its maximal irreducible Irreducible SFT's are easier to understand than general SFT's, and \) The orbit of \(x \in X\) is the trajectory \(\\) is dense in \(X\. In broad terms, an invertible dynamical system is a set \(X\ ,\) together with an invertible mapping \(T:X \rightarrow X\. 5 Invariants of conjugacy and variants of the conjugacy problem.4 Shifts of finite type and sofic shifts.
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