scholarly journals A Juzvinskii addition theorem for finitely generated free group actions

2012 ◽  
Vol 34 (1) ◽  
pp. 95-109 ◽  
Author(s):  
LEWIS BOWEN ◽  
YONATAN GUTMAN
Keyword(s):  
Markov Processes ◽  
Free Group ◽  
Group Actions ◽  
Addition Theorem ◽  
Skew Product ◽  
Compact Lie Groups ◽  
Finitely Generated ◽  
Skew Products ◽  
Totally Disconnected ◽  
Free Group Actions

AbstractThe classical Juzvinskii addition theorem states that the entropy of an automorphism of a compact group decomposes along invariant subgroups. Thomas generalized the theorem to a skew-product setting. Using L. Bowen’s f-invariant, we prove the addition theorem for actions of finitely generated free groups on skew-products with compact totally disconnected groups or compact Lie groups (correcting an error in L. Bowen [Nonabelian free group actions: Markov processes, the Abramov–Rohlin formula and Yuzvinskii’s formula. Ergod. Th. & Dynam. Sys.30(6) (2010), 1629–1663]) and discuss examples.

Non-abelian free group actions: Markov processes, the Abramov–Rohlin formula and Yuzvinskii’s formula

2009 ◽  
Vol 30 (6) ◽  
pp. 1629-1663 ◽  
Author(s):  
LEWIS BOWEN
Keyword(s):  
Markov Chain ◽  
Markov Chains ◽  
Markov Processes ◽  
Free Group ◽  
Skew Product ◽  
Addition Formula ◽  
Transition Matrices ◽  
Classical Formula ◽  
Algebraic Actions ◽  
Free Group Actions

AbstractThis paper introduces Markov chains and processes over non-abelian free groups and semigroups. We prove a formula for the f-invariant of a Markov chain over a free group in terms of transition matrices that parallels the classical formula for the entropy a Markov chain. Applications include free group analogues of the Abramov–Rohlin formula for skew-product actions and Yuzvinskii’s addition formula for algebraic actions.

scholarly journals A subgroup formula for f-invariant entropy

2012 ◽  
Vol 34 (1) ◽  
pp. 263-298 ◽  
Author(s):  
BRANDON SEWARD
Keyword(s):  
Free Group ◽  
Free Groups ◽  
Amenable Group ◽  
Group Actions ◽  
Finitely Generated ◽  
Definition Of ◽  
Sofic Groups ◽  
Free Group Action ◽  
Measure Entropy ◽  
Free Group Actions

AbstractWe study a measure entropy for finitely generated free group actions called f-invariant entropy. The f-invariant entropy was developed by L. Bowen and is essentially a special case of his measure entropy theory for actions of sofic groups. In this paper we relate the f-invariant entropy of a finitely generated free group action to the f-invariant entropy of the restricted action of a subgroup. We show that the ratio of these entropies equals the index of the subgroup. This generalizes a well-known formula for the Kolmogorov–Sinai entropy of amenable group actions. We then extend the definition of f-invariant entropy to actions of finitely generated virtually free groups. We also obtain a numerical virtual measure conjugacy invariant for actions of finitely generated virtually free groups.

Rings with Fixed-Point-Free Group Actions

1973 ◽  
Vol s3-27 (1) ◽  
pp. 69-87 ◽  
Author(s):  
G. M. Bergman ◽  
I. M. Isaacs
Keyword(s):  
Fixed Point ◽  
Free Group ◽  
Group Actions ◽  
Free Group Actions

scholarly journals Scale functions and tree ends

2001 ◽  
Vol 70 (2) ◽  
pp. 273-292 ◽  
Author(s):  
A. Kepert ◽  
G. Willis
Keyword(s):  
Free Group ◽  
Reduced Form ◽  
Scale Function ◽  
Direct Products ◽  
Finitely Generated ◽  
Index Set ◽  
Scale Functions ◽  
Totally Disconnected

AbstractA class of totally disconnected groups consisting of partial direct products on an index set is examined. For such a group, the scale function is found, and for automorphisms arising from permutations of the index set, the tidy subgroups are characterised. When applied to the case where the index set is a finitely-generated free group and the permutation is translation by an element x of the group, the scale depends on the cyclically reduced form of x and the tidy subgroup on the element which conjugates x to its cyclically reduced form.

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