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Papers by Frédéric Mathéus

We give a geometric description of the Poisson boundaries of certain extensions of free and hyper... more We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic groups. We rely upon the description of Poisson boundaries by means of a topological compactification as developed by Kaimanovich. All the groups studied here share the property of admitting a sufficiently complicated action on some real tree.

Let G be a free product of a finite family of finite groups, with the set of generators being for... more Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy, and the volume of the group.

This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mair... more This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mairesse and F.Math\'eus. It contains the details of the computations and the proofs of the results concerning the examples treated there.

International Journal of Algebra and Computation, 2006

We consider the Artin groups of dihedral type Ak =h a; bj prod(a; b; k) = prod(b; a; k)i where pr... more We consider the Artin groups of dihedral type Ak =h a; bj prod(a; b; k) = prod(b; a; k)i where prod(s; t; k) = ststs:::, with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of Ak with respect to the Artin generatorsfa; b; a 1; b 1g

We consider a transient nearest neighbor random walk on a group G with finite set of generators �... more We consider a transient nearest neighbor random walk on a group G with finite set of generators �. The pair (G,�) is assumed to admit a nat- ural notion of normal form words which are modified only locally when multiplied by generators. The basic examples are the free products of a finitely generated free group and a finite family of

Journal of The London Mathematical Society-second Series, 2007

Let G be a free product of a finite family of finite groups, with the set of generators being for... more Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy, and the volume of the group.

This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mair... more This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mairesse and F.Math\'eus. It contains the details of the computations and the proofs of the results concerning the examples treated there.

This paper is an appendix to the paper ``Randomly Growing Braid on Three Strands and the Manta Ra... more This paper is an appendix to the paper ``Randomly Growing Braid on Three Strands and the Manta Ray'' by J. Mairesse and F. Math\'eus (to appear in the Annals of Applied Probability). It contains the details of some computations, and the proofs of some results concerning the examples treated there, as well as some extensions.

Annals of Applied Probability, 2007

In memory of Daniel Mollier, our former mathematics teacher at the Lycée Louis le Grand, Paris.

We give a geometric description of the Poisson boundaries of certain extensions of free and hyper... more We give a geometric description of the Poisson boundaries of certain extensions of free and hyperbolic groups. In particular, we get a full description of the Poisson boundaries of free-by-cyclic groups. We rely upon the description of Poisson boundaries by means of a topological compactification as developed by Kaimanovich. All the groups studied here share the property of admitting a sufficiently complicated action on some real tree.

Let G be a free product of a finite family of finite groups, with the set of generators being for... more Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy, and the volume of the group.

This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mair... more This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mairesse and F.Math\'eus. It contains the details of the computations and the proofs of the results concerning the examples treated there.

International Journal of Algebra and Computation, 2006

We consider the Artin groups of dihedral type Ak =h a; bj prod(a; b; k) = prod(b; a; k)i where pr... more We consider the Artin groups of dihedral type Ak =h a; bj prod(a; b; k) = prod(b; a; k)i where prod(s; t; k) = ststs:::, with k terms in the product on the right-hand side. We prove that the spherical growth series and the geodesic growth series of Ak with respect to the Artin generatorsfa; b; a 1; b 1g

We consider a transient nearest neighbor random walk on a group G with finite set of generators �... more We consider a transient nearest neighbor random walk on a group G with finite set of generators �. The pair (G,�) is assumed to admit a nat- ural notion of normal form words which are modified only locally when multiplied by generators. The basic examples are the free products of a finitely generated free group and a finite family of

Journal of The London Mathematical Society-second Series, 2007

Let G be a free product of a finite family of finite groups, with the set of generators being for... more Let G be a free product of a finite family of finite groups, with the set of generators being formed by the union of the finite groups. We consider a transient nearest-neighbour random walk on G. We give a new proof of the fact that the harmonic measure is a special Markovian measure entirely determined by a finite set of polynomial equations. We show that in several simple cases of interest, the polynomial equations can be explicitely solved, to get closed form formulas for the drift. The examples considered are the modular group Z/2Z*Z/3Z, Z/3Z*Z/3Z, Z/kZ*Z/kZ, and the Hecke groups Z/2Z*Z/kZ. We also use these various examples to study Vershik's notion of extremal generators, which is based on the relation between the drift, the entropy, and the volume of the group.

This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mair... more This paper is an appendix to the paper "Random walks on free products of cyclic groups" by J.Mairesse and F.Math\'eus. It contains the details of the computations and the proofs of the results concerning the examples treated there.

This paper is an appendix to the paper ``Randomly Growing Braid on Three Strands and the Manta Ra... more This paper is an appendix to the paper ``Randomly Growing Braid on Three Strands and the Manta Ray'' by J. Mairesse and F. Math\'eus (to appear in the Annals of Applied Probability). It contains the details of some computations, and the proofs of some results concerning the examples treated there, as well as some extensions.

Annals of Applied Probability, 2007

In memory of Daniel Mollier, our former mathematics teacher at the Lycée Louis le Grand, Paris.