Let
F
k
{F_k}
denote the free group on
k
k
generators,
1
>
k
>
∞
1 > k > \infty
, and let
S
S
denote a set of free generators and their inverses. Define
σ
n
=
d
1
#
S
n
Σ
w
∈
S
n
w
{\sigma _n} \stackrel {d}{=} \frac {1}{{\# {S_n}}}{\Sigma _{w \in {S_n}}}w
, where
S
n
=
{
w
:
|
w
|
=
n
}
{S_n} = \{ w:|w| = n\}
, and
|
⋅
|
| \cdot |
denotes the word length on
F
k
{F_k}
induced by
S
S
. Let
(
X
,
B
,
m
)
(X, \mathcal {B}, m)
be a probability space on which
F
k
{F_k}
acts ergodically by measure preserving transformations. We prove a pointwise ergodic theorem for the sequence of operators
σ
n
′
=
1
2
(
σ
n
+
σ
n
+
1
)
\sigma _n^\prime = \frac {1}{2}({\sigma _n} + {\sigma _{n + 1}})
acting on
L
2
(
X
)
{L^2}(X)
, namely:
σ
n
′
f
(
x
)
→
∫
X
f
d
m
\sigma _n^\prime f(x) \to \int _X {f\,dm}
almost everywhere, for each
f
f
in
L
2
(
X
)
{L^2}(X)
. We also show that the sequence
σ
2
n
{\sigma _{2n}}
converges to a conditional expectation operator with respect to a
σ
\sigma
-algebra which is invariant under
F
k
{F_k}
. The proof is based on the spectral theory of the (commutative) convolution subalgebra of
ℓ
1
(
F
k
)
{\ell ^1}({F_k})
generated by the elements
σ
n
,
n
≥
0
{\sigma _n},\,\;n \geq 0
. We then generalize the discussion to algebras arising as a Gelfand pair associated with the group of automorphisms
G
(
r
1
,
r
2
)
G({r_1},\;{r_2})
of a semi-homogeneous tree
T
(
r
1
,
r
2
)
T({r_1},\;{r_2})
, where
r
1
≥
2
,
r
2
≥
2
,
r
1
+
r
2
>
4
{r_1} \geq 2,\;{r_2} \geq 2,\;{r_1} + {r_2} > 4
. (The case of
F
k
{F_k}
corresponds to that of a homogeneous tree of valency
2
k
2k
.) We prove similar pointwise ergodic theorems for two classes of subgroups of
G
(
r
1
,
r
2
)
G({r_1},\;{r_2})
. One is the class of closed noncompact boundary-transitive subgroups, including any simple algebraic group of split rank one over a local field, for example,
P
S
L
2
(
Q
p
)
PS{L_2}({\mathbb {Q}_p})
. The second class is that of lattices complementing a maximal compact subgroup. We also prove a strong maximal inequality in
L
2
(
X
)
{L^2}(X)
for the groups listed above, as well as a mean ergodic theorem for unitary representations of the groups (due to
Y
{\text {Y}}
. Guivarc’h for
F
k
{F_k}
). Finally, we describe the structure and spectral theory of a noncommutative algebra which arises naturally in the present context, namely the double coset algebra associated with the subgroup of
G
(
r
1
,
r
2
)
G({r_1},\;{r_2})
stabilizing a geometric edge. The results are applied to prove mean ergodic theorems for a family of lattices in
G
(
r
1
,
r
2
)
G({r_1},\;{r_2})
, which includes, for example,
P
S
L
2
(
Z
)
PS{L_2}(\mathbb {Z})
.