The geometries of the Freudenthal-Tits magic square We show that in case of Cayley graphs of Coxeter groups, one can fully characterise the discrete ones in terms of the symmetries of the corresponding Coxeter system. It therefore makes sense to ask for which graphs is the topology not discrete. Group of automorphisms of a connected locally finite graph is naturally a totally disconnected locally compact topological group, when equipped with the permutation topology. We relate these two subjects!Īutomorphism groups of Cayley graphs of Coxeter groups: when are they discrete? Sandpile models are about how things spread along a grid (think of Covid!) and Leavitt algebras are algebras associated to graphs. Our results generalise the phase transition on the Toeplitz noncommutative tori used as building blocks in work of Brownlowe, Hawkins and Sims, and of Afsar, an Huef, Raeburn and Sims. These in turn correspond to probability measures on a classical torus whose dimension depends on a certain degeneracy index of the restricted cocycle. Laca, in which we show that the equilibrium states of the resulting C*-dynamical system are parametrised by tracial states of the noncommutative torus corresponding to the restriction of the cocycle to the vanishing coordinates of $r$. Moreover, not that the groups studied by Serre correspond to the case where the ideal $I$ coincides with the ring $R$.ĭiscrete two-generator subgroups of $PSL(2,\mathbb^n$ twisted by a $2$-cocycle is a Toeplitz extension of an $n$-dimensional noncommutative torus, on which each vector $r \in [0,\infty)^n$ determines a one-parameter subgroup of the gauge action. These groups play, in the function field context, the same role as the Hecke Congruence subgroups of $SL_2(Z)$. Then By, using Bass-Serre theory, we describe the combinatorial structure of $H$. More specifically, we give an explicit formula for the cusp number $H\backslash T$. In this exposition we describe the quotient graph $H\backslash T$ defined from the action on $T$ of the group $H$ consisting of matrices that are upper triangular modulo $I$, where $I$ is an ideal $R$. It is not hard to see that finite index subgroups inherit this property. In particular, Serre shows that $GL_2(R)\backslash T$ is the union of a finite graph and a finite number of ray shaped subgraphs, which are called cusps. In order to study group of the form $GL_2(R)$, Serre describes the quotient graph $GL_2(R)\backslash T$, where $T$ is the Bruhat-Tits tree defined from $SL_2(K)$. For each closed point $P_\infty$ of $C$, let $R$ be the ring of functions that are regular outside $P_\infty$, and let $K$ be the completion path $P_\infty$ of the function field of $C$. Let $C$ be a smooth, projective, and geometrically connected curve defined over a finite field $F$. Quotients of the Bruhat-Tits tree function field analogs of the Hecke congruence subgroups 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.
Toeplitz subshift conjugacy invariant series#
David Ruelle, Thermodynamic formalism, Encyclopedia of Mathematics and its Applications, vol. 5, Addison-Wesley Publishing Co., Reading, Mass., 1978.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.353 (2001), 3713-3739Ībstract: For a dynamical system $(X,T)$ and function $f:X\to \mathbb $ action satisfying expansiveness and specification, Trans. Rotation, entropy, and equilibrium states
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