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Cogrowth of Arbitrary Graphs
"... Abstract. A “cogrowth set ” of a graph G is the set of vertices in the universal cover of G which are mapped by the universal covering map onto a given vertex of G. Roughly speaking, a cogrowth set is large if and only if G is small. In particular, when G is regular, a cogrowth constant (a measure o ..."
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Abstract. A “cogrowth set ” of a graph G is the set of vertices in the universal cover of G which are mapped by the universal covering map onto a given vertex of G. Roughly speaking, a cogrowth set is large if and only if G is small. In particular, when G is regular, a cogrowth constant (a measure
Cogrowth of regular graphs
 Proc. Amer. Math. Soc
, 1992
"... Abstract. Let & be a ¿regular graph and T the covering tree of S. We define a cogrowth constant of & in T and express it in terms of the first eigenvalue of the Laplacian on S. As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue of ..."
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Cited by 12 (3 self)
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Abstract. Let & be a ¿regular graph and T the covering tree of S. We define a cogrowth constant of & in T and express it in terms of the first eigenvalue of the Laplacian on S. As a corollary, we show that the cogrowth constant is as large as possible if and only if the first eigenvalue
On rationality of the cogrowth series
 Proc. Amer. Math. Soc
, 1998
"... Abstract. The cogrowth series of a group G depends on the presentation of the group. We show that the cogrowth series of a nonempty presentation is a rational function not equal to 1 if and only if G is finite. Except for the trivial group, this property is independent of presentation. ..."
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Cited by 7 (0 self)
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Abstract. The cogrowth series of a group G depends on the presentation of the group. We show that the cogrowth series of a nonempty presentation is a rational function not equal to 1 if and only if G is finite. Except for the trivial group, this property is independent of presentation.
On the cogrowth of Thompson’s group
 F . Groups Complex. Cryptol
"... We investigate the cogrowth and distribution of geodesics in R. Thompson’s group F. 1 ..."
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Cited by 5 (3 self)
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We investigate the cogrowth and distribution of geodesics in R. Thompson’s group F. 1
COGROWTH AND ESSENTIALITY IN GROUPS AND ALGEBRAS
, 1993
"... The cogrowth of a subgroup is defined as the growth of a set of coset representatives which are of minimal length. A subgroup is essential if it intersects nontrivially every nontrivial subgroup. The main result of this paper is that every function f: N ∪ {0}−→N which is strictly increasing, but a ..."
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The cogrowth of a subgroup is defined as the growth of a set of coset representatives which are of minimal length. A subgroup is essential if it intersects nontrivially every nontrivial subgroup. The main result of this paper is that every function f: N ∪ {0}−→N which is strictly increasing
QUASIREGULAR GRAPHS, COGROWTH, AND AMENABILITY
"... Abstract. We extend Grigrochuk’s cogrowth criterion for amenability of groups to the case of nonregular graphs for which a certain regularity condition is satisfied. The proof involves generalized Laplacians which are inverses of growth series and whose determinants are closely related to zeta func ..."
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Abstract. We extend Grigrochuk’s cogrowth criterion for amenability of groups to the case of nonregular graphs for which a certain regularity condition is satisfied. The proof involves generalized Laplacians which are inverses of growth series and whose determinants are closely related to zeta
Nonbacktracking random walks and cogrowth of graphs
 Canadian Journal of Mathematics
"... Abstract. Let X be a locally finite, connected graph without vertices of degree 1. Nonbacktracking random walk moves at each step with equal probability to one of the “forward ” neighbours of the actual state, i.e., it does not go back along the preceding edge to the preceding state. This is not a ..."
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Cited by 16 (1 self)
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concerning cogrowth of groups, and makes the extension of that result to arbitrary regular graphs rigorous. Even when X is nonregular, but small cycles are dense in X, we show that the graph X is nonamenable if and only if the nonbacktracking nstep transition probabilities decay exponentially fast
On the cogrowth of Thompson’s group F 1
"... We investigate the cogrowth and distribution of geodesics in R. Thompson’s group F. ..."
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We investigate the cogrowth and distribution of geodesics in R. Thompson’s group F.
The
"... ratio and generating function of cogrowth coefficients of finitely generated groups ..."
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ratio and generating function of cogrowth coefficients of finitely generated groups
The
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"... ratio and generating function of cogrowth coefficients of finitely generated groups ..."
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ratio and generating function of cogrowth coefficients of finitely generated groups
Results 1  10
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