# Groups, Geometry, and Dynamics

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**Volume 4, Issue 2, 2010, pp. 209–249**

**DOI: 10.4171/GGD/81**

Published online: 2010-02-21

Representation zeta functions of wreath products with finite groups

Laurent Bartholdi^{[1]}and Pierre de la Harpe

^{[2]}(1) Georg-August-Universität Göttingen, Germany

(2) Université de Genève, Switzerland

Let `G` be a group which has a finite number `r`_{n}(`G`)
of
irreducible
linear
representations
in
GL_{n}(ℂ) for
all `n` ≥ 1.`ζ`(`G`,`s`) = ∑`n` = 1`r`_{n}(`G`)`n`^{−s}
be its representation zeta function. First, in case `G` = `H`
≀_{X} `Q` is a permutational wreath
product
with respect to a permutation group `Q` on a finite set `X`,
we
establish
a
formula
for
`ζ`(`G`,`s`) in
terms of
the zeta functions of `H` and of subgroups of `Q`,
and of the Möbius function associated to the lattice Π_{Q}(`X`)
of
partitions
of
`X` in orbits under subgroups of `Q`.

Then we consider groups
`W`(`Q`,`k`) = (··· (`Q`
≀_{X} `Q`) ≀_{X} `Q`
···) ≀_{X} `Q`
which are iterated wreath products (with `k` factors `Q`),
and
several
related
infinite
groups
`W`(`Q`),
including the profinite group lim_{k}`W`(`Q`,`k`),
a
locally
finite
group
lim_{k}`W`(`Q`,`k`),
and
several
finitely
generated dense subgroups of lim_{k}`W`(`Q`,`k`).
Under
convenient
hypotheses
(in
particular
`Q` should be perfect),
we show that `r`_{n}(`W`(`Q`))
<
∞
for
all
`n` ≥ 1,
and we establish that the Dirichlet series `ζ`(`W`(`Q`),`s`)
has
a
finite
and
positive
abscissa
of
convergence
`σ`_{0} = `σ`_{0}(`W`(`Q`)).
Moreover,
the
function
`ζ`(`W`(`Q`),`s`)
satisfies a remarkable functional equation
involving `ζ`(`W`(`Q`),`es`) for `e`
∈ {1,…,d},
where `d` = |`X`|.
As a consequence of this,
we exhibit some properties of the function,
in particular that `ζ`(`W`(`Q`),`s`)
has
a
root-type
singularity
at
`σ`_{0},
with a finite value at `σ`_{0}
and a Puiseux expansion around `σ`_{0}.

We finally report some numerical computations
for `Q` = `A`_{5} and `Q` = PGL_{3}(**F**_{2}).

*Keywords: *Irreducible linear representations, finite groups, wreath products, `d`-ary tree, groups of automorphisms of rooted trees, Clifford theory, Dirichlet series, representation zeta function

Bartholdi Laurent, de la Harpe Pierre: Representation zeta functions of wreath products with finite groups. *Groups Geom. Dyn.* 4 (2010), 209-249. doi: 10.4171/GGD/81