Session S28 - Knots, Surfaces, 3-manifolds

Thursday, July 15, 17:20 ~ 17:50 UTC-3

Dynamical systems on hyperbolic groups

Yo'av Rieck

University of Arkansas , USA   -   This email address is being protected from spambots. You need JavaScript enabled to view it.

Let $G$ be an infinite hyperbolic group.

By a \em dynamical system \em on $G$ we mean an action of $G$ on a compact space $X$. The most commonly studied (and best understood) type of dynamical system, called SFT (subshift of finite type), is given by a closed, $G$-invariant subspace $X \subset A^G$, where $A$ is any finite set. We will explain these terms in the talk and show why an SFT on $G$ is essentially a ``tiling'' of $G$.

Gromov studied SFT's on $G$ in his original paper about hyperbolic groups, and much work on the subject was done by Coornaert and Papadopoulos. In particular, $G$ admits an SFT that can be used to study its action on its boundary.

A non-empty SFT is called \em strongly aperiodic \em if the stabilizer of every point is trivial. The question of which finitely generated, infinite groups admits a strongly aperiodic SFT has a long history, dating back to the foundational work of Wang and Berger in the 60's. Few groups are known not to admit one, and many are known to admit one; however, until the current work, the only hyperbolic groups that were known to admit a strongly aperiodic SFT were surface groups (Cohen and Goodman-Strauss).

In this talk we will describe the construction of a strongly aperiodic SFT when $G$ is one-ended, which is the key for the following result:

{\bf Theorem.} An infinite hyperbolic group admits a strongly aperiodic SFT if and only if it is one-ended.

Time permitting we will discuss further development related to this construction.

Joint work with David Bruce Cohen and Chaim Goodman--Strauss.

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