arXiv subjects

Laurent Bartholdi

Publications and source records attributed to Laurent Bartholdi.

114 records · Page 7Linked to original sources

arXiv2000

Sous-groupes paraboliques et representations de groupes branches

Let G be a branch group (as defined by Grigorchuk) acting on a tree T. A parabolic subgroup P is the stabiliser of an infinite geodesic ray in T. We denote by $ρ_{G/P}$ the associated quasi-regular representation. If G is discrete, these representations are irreducible, but if G is profinite, they split as a direct sum of finite-dimensionalrepresentations $ρ_{G/P_{n+1}}\ominusρ_{G/P_n}$, where P_n is the stabiliser of a level-n vertex in T. For a few concrete examples, we completely split $ρ_{G/P_n}$ in irreducible components. $(G,P_n)$ and $(G,P)$ are Gelfand pairs, whence new occurrences of abelian Hecke algebra.

math.GR↗

arXiv2000

Growth series of some hyperbolic graphs and Salem numbers

Extending the analogous result of Cannon and Wagreich for the fundamental groups of surfaces, we show that, for the l-regular graphs X associated to regular tessellations of hyperbolic plane by m-gons, the denominators of the growth series (which are rational and were computed by Floyd and Plotnick) are reciprocal Salem polynomials. As a consequence, the growth rates of these graphs are Salem numbers. We then derive some regularity properties for the coefficients $a_n$ of the growth series: they satisfy $$Kλ^n-R 0$, $λ>1$.

math.GR↗

arXiv2000

The Growth of Grigorchuk's Group

In 1980 Rostislav Grigorchuk constructed a group $G$ of intermediate growth, and later obtained the following estimates on its growth function: $$e^{\sqrt{n}}\precsimγ(n)\precsim e^{n^β},$$ where $β=\log_{32}(31)\approx0.991$. Using elementary methods we bring the upper bound down to $\log(2)/\log(2/η)\approx0.767$, where $η\approx0.811$ is the real root of the polynomial $X^3+X^2+X-2$.

math.GR↗

arXiv2000

On the Word and Period Growth of some Groups of Tree Automorphisms

We generalize a class of groups defined by Rostislav Grigorchuk to a much larger class of groups, and provide upper and lower bounds for their word growth (they are all of intermediate growth) and period growth (under a small additional condition, they are periodic).

math.GR↗

arXiv1999

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

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 approximation'' method. These spectra are simple transforms of the Julia sets associated to some quadratic maps. The graphs involved are Schreier graphs of fractal groups of intermediate growth, and are also ``substitutional graphs''. We also formulate our results in terms of Hecke type operators related to some irreducible quasi-regular representations of fractal groups and in terms of the Markovian operator associated to noncommutative dynamical systems via which these fractal groups were originally defined. In the computations we performed, the self-similarity of the groups is reflected in the self-similarity of some operators; they are approximated by finite counterparts whose spectrum is computed by an ad hoc factorization process.

math.GR↗

arXiv1999

Lamps, Factorizations and Finite Fields

I answer a question from the 1993 International Mathematical Olympiads by constructing an equivalent algebraic problem, and unearth a surprising behaviour of some polynomials over the two-element field.

math.CO↗