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Counting Paths in Graphs

- L. Bartholdi
- Mathematics
- 18 December 2000

We give a simple combinatorial proof of a formula that extends a result by Grigorchuk (rediscovered by Cohen) relating cogrowth and spectral radius of random walks. Our main result is an explicit… Expand

Amenability via random walks

- L. Bartholdi, B. Virág
- Mathematics
- 19 May 2003

We use random walks to show that the Basilica group is amenable, answering an open question of Grigorchuk and ˙ Zuk. Our results separate the class of amenable groups from the closure of… Expand

Thurston equivalence of topological polynomials

- L. Bartholdi, V. Nekrashevych
- Mathematics
- 4 October 2005

We answer Hubbard's question on determining the Thurston equivalence class of “twisted rabbits”, i.e. composita of the “rabbit” polynomial with nth powers of the Dehn twists about its ears. The… Expand

Horocyclic products of trees

- L. Bartholdi, M. Neuhauser, W. Woess
- Mathematics
- 17 January 2006

Let $T_1,\dots, T_d$ be homogeneous trees with degrees $q_1+1, \dots, q_d+1 \ge 3,$ respectively. For each tree, let $\hor:T_j \to \Z$ be the Busemann function with respect to a fixed boundary point… Expand

From fractal groups to fractal sets

- L. Bartholdi, R. Grigorchuk, V. Nekrashevych
- Mathematics
- 1 February 2002

The idea of self-similarity is one of the most fundamental in the modern mathematics. The notion of “renormalization group”, which plays an essential role in quantum field theory, statistical physics… Expand

On the Spectrum of Hecke Type Operators related to some Fractal Groups

- L. Bartholdi, R. Grigorchuk
- Mathematics
- 19 October 1999

We give the first example of a connected 4-regular graph whose Laplace operator's spectrum is a Cantor set, as well as several other computations of spectra following a common ``finite… Expand

Groups and Lie algebras corresponding to the Yang–Baxter equations

- L. Bartholdi, B. Enriquez, P. Etingof, E. Rains
- Mathematics
- 28 September 2005

On Parabolic Subgroups and Hecke Algebras of Some Fractal Groups

- L. Bartholdi, R. Grigorchuk
- Mathematics
- 25 November 1999

We study the subgroup structure, Hecke algebras, quasi-regular representations, and asymptotic properties of some fractal groups of branch type. We introduce parabolic subgroups, show that they are… Expand

On amenability of automata groups

- L. Bartholdi, V. Kaimanovich, V. Nekrashevych
- Mathematics
- 20 February 2008

We show that the group of bounded automatic automorphisms of a rooted tree is amenable, which implies amenability of numerous classes of groups generated by finite automata. The proof is based on… Expand

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