New York Journal of Mathematics
Volume 21 (2015) 811-822


David Carroll and Andrew Penland

Periodic points on shifts of finite type and commensurability invariants of groups

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Published: August 25, 2015
Keywords: Symbolic dyamics, shifts of finite type
Subject: 37B10, 37B50, 37C25, 52C23

We explore the relationship between subgroups and the possible shifts of finite type (SFTs) which can be defined on a group. In particular, we investigate two group invariants, weak periodicity and strong periodicity, defined via symbolic dynamics on the group. We show that these properties are invariants of commensurability. Thus, many known results about periodic points in SFTs defined over groups are actually results about entire commensurability classes. Additionally, we show that the property of being not strongly periodic (i.e., the property of having a weakly aperiodic SFT) is preserved under extensions with finitely generated kernels. We conclude by raising questions and conjectures about the relationship of these invariants to the geometric notions of quasi-isometry and growth.

Author information

David Carroll:
Dept. Of Mathematics, Mailstop 3368, Texas A&M University, College Station, TX 77843-3368, USA

Andrew Penland:
Dept. of Mathematics and Computer Science, Western Carolina University, Stillwell 426, Cullowhee, NC 28723, USA