## View abstract

### 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. document.getElementById('cloakac46e87c8d7b1707e81c55699aa0f4d8').innerHTML = ''; var prefix = '&#109;a' + 'i&#108;' + '&#116;o'; var path = 'hr' + 'ef' + '='; var addyac46e87c8d7b1707e81c55699aa0f4d8 = 'y&#111;&#97;v' + '&#64;'; addyac46e87c8d7b1707e81c55699aa0f4d8 = addyac46e87c8d7b1707e81c55699aa0f4d8 + '&#117;&#97;rk' + '&#46;' + '&#101;d&#117;'; var addy_textac46e87c8d7b1707e81c55699aa0f4d8 = 'y&#111;&#97;v' + '&#64;' + '&#117;&#97;rk' + '&#46;' + '&#101;d&#117;';document.getElementById('cloakac46e87c8d7b1707e81c55699aa0f4d8').innerHTML += '<a ' + path + '\'' + prefix + ':' + addyac46e87c8d7b1707e81c55699aa0f4d8 + '\'>'+addy_textac46e87c8d7b1707e81c55699aa0f4d8+'<\/a>';

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.