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.

GAUGE SYMMETRY BREAKING IN SUPERCONFORMAL ORBIFOLDS

1991 ◽  
Vol 06 (07) ◽  
pp. 591-603
Author(s):  
B.R. GREENE ◽  
M.R. PLESSER ◽  
EDMOND RUSJAN ◽  
XING-MIN WANG
Keyword(s):  
Gauge Symmetry ◽  
Symmetry Breaking ◽  
Symmetry Group ◽  
Free Group ◽  
Group Actions ◽  
Wilson Loops ◽  
String Compactifications ◽  
Free Group Actions

We study the construction of (2, 0) theories from orbifolds of N=2 minimal superconformal string compactifications with non-trivial Wilson loops. In particular, we exploit the connection between geometrical and exactly soluble string vacua to arrive at a mean of analyzing Calabi-Yau orbifolds containing ‘Wilson loops’ associated with non-free group actions, breaking the E6 gauge symmetry of the model as well the ‘shadow’ E8 gauge symmetry group. We apply our results to recently proposed three generation constructions of this sort and find spectra which differ from previous claims and which possess exceptionally desirable phenomenological properties.

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