Abstract
For a locally compact group 𝐺, we write
S
U
B
(
G
)
{\mathcal{SUB}}(G)
for the space of closed subgroups of 𝐺 endowed with the Chabauty topology.
For any positive integer 𝑛, we associate to 𝐺 the function
δ
G
,
n
\delta_{G,n}
from
G
n
G^{n}
to
S
U
B
(
G
)
{\mathcal{SUB}}(G)
defined by
δ
G
,
n
(
g
1
,
…
,
g
n
)
=
gp
¯
(
g
1
,
…
,
g
n
)
,
\delta_{G,n}(g_{1},\ldots,g_{n})=\overline{\mathrm{gp}}(g_{1},\ldots,g_{n}),
where
gp
¯
(
g
1
,
…
,
g
n
)
\overline{\mathrm{gp}}(g_{1},\ldots,g_{n})
denotes the closed subgroup topologically generated by
g
1
,
…
,
g
n
g_{1},\ldots,g_{n}
.
It would be interesting to know for which groups 𝐺 the function
δ
G
,
n
\delta_{G,n}
is continuous for every 𝑛.
Let
[
HW
]
[\mathtt{HW}]
be the class of such groups.
Some interesting properties of the class
[
HW
]
[\mathtt{HW}]
are established.
In particular, we prove that
[
HW
]
[\mathtt{HW}]
is properly included in the class of totally disconnected locally compact groups.
The class of totally disconnected locally compact locally pronilpotent groups is included in
[
HW
]
[\mathtt{HW}]
.
Also, we give an example of a solvable totally disconnected locally compact group not contained in
[
HW
]
[\mathtt{HW}]
.