The Spectra of Lamplighter Groups and Cayley Machines
نویسندگان
چکیده
We calculate the spectra and spectral measures associated to random walks on restricted wreath products G wr Z, with G a finite abelian group, by realizing them as groups generated by automata. This generalizes the work of Grigorchuk and Żuk on the lamplighter group. More generally we calculate the spectra of random walks on groups generated by Cayley machines of finite groups and calculate Kesten-von Neumann-Serre spectral measures of random walks on their Schreier graphs with respect to their parabolic subgroups. Some of these results were obtained by Dicks and Schick via a different method.
منابع مشابه
Cross-wired lamplighter groups and linearity of automata groups
We consider the two generalizations of lamplighter groups: automata groups generated by Cayley machines and cross-wired lamplighter groups. For a finite step two nilpotent group with central squares, we study its associated Cayley machine and give a presentation of the corresponding automata group. We show the automata group is a crosswired lamplighter group and does not embed in the wreath pro...
متن کاملThe Sigma Invariants of the Lamplighter Groups
We compute the Bieri-Neumann-Strebel-Renz geometric invariants, Σn, of the lamplighter groups Lm by using the Diestel-Leader graph DL(m,m) to represent the Cayley graph of Lm.
متن کاملDead End Words in Lamplighter Groups and Other Wreath Products Sean Cleary and Jennifer Taback
We explore the geometry of the Cayley graphs of the lamplighter groups and a wide range of wreath products. We show that these groups have dead end elements of arbitrary depth with respect to their natural generating sets. An element w in a group G with finite generating set X is a dead end element if no geodesic ray from the identity to w in the Cayley graph Γ(G, X) can be extended past w. Add...
متن کاملMeasuring Closeness Between Cayley Automatic Groups and Automatic Groups
In this paper we introduce a way to estimate a level of closeness of Cayley automatic groups to the class of automatic groups using a certain numerical characteristic. We characterize Cayley automatic groups which are not automatic in terms of this numerical characteristic and then study it for the lamplighter group, the Baumslag–Solitar groups and the Heisenberg group.
متن کاملGreen kernel estimates and the full Martin boundary for random walks on lamplighter groups and Diestel–Leader graphs
We determine the precise asymptotic behaviour (in space) of the Green kernel of simple random walk with drift on the Diestel–Leader graph DL(q, r), where q, r 2. The latter is the horocyclic product of two homogeneous trees with respective degrees q + 1 and r + 1. When q = r , it is the Cayley graph of the wreath product (lamplighter group) Zq Z with respect to a natural set of generators. We d...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2008