We study singularity properties of word maps on semisimple Lie algebras, semisimple algebraic groups and matrix algebras and obtain various applications to random walks induced by word measures on compact
p
p
-adic groups.
Given a word
w
w
in a free Lie algebra
L
r
\mathcal {L}_{r}
, it induces a word map
φ
w
:
g
r
→
g
\varphi _{w}:\mathfrak {g}^{r}\rightarrow \mathfrak {g}
for every semisimple Lie algebra
g
\mathfrak {g}
. Given two words
w
1
∈
L
r
1
w_{1}\in \mathcal {L}_{r_{1}}
and
w
2
∈
L
r
2
w_{2}\in \mathcal {L}_{r_{2}}
, we define and study the convolution of the corresponding word maps
φ
w
1
∗
φ
w
2
≔
φ
w
1
+
φ
w
2
:
g
r
1
+
r
2
→
g
\varphi _{w_{1}}*\varphi _{w_{2}}≔\varphi _{w_{1}}+\varphi _{w_{2}}:\mathfrak {g}^{r_{1}+r_{2}}\rightarrow \mathfrak {g}
.
By introducing new degeneration techniques, we show that for any word
w
∈
L
r
w\in \mathcal {L}_{r}
of degree
d
d
, and any simple Lie algebra
g
\mathfrak {g}
with
φ
w
(
g
r
)
≠
0
\varphi _{w}(\mathfrak {g}^{r})\neq 0
, one obtains a flat morphism with reduced fibers of rational singularities (abbreviated an (FRS) morphism) after taking
O
(
d
4
)
O(d^{4})
self-convolutions of
φ
w
\varphi _{w}
. Similar results are obtained for matrix word maps. We deduce that a group word map of length
ℓ
\ell
becomes (FRS), locally around identity, after
O
(
ℓ
4
)
O(\ell ^{4})
self-convolutions, for every semisimple algebraic group
G
_
\underline {G}
. We furthermore provide uniform lower bounds on the log canonical threshold of the fibers of Lie algebra, matrix and group word maps. For the commutator word
w
0
=
[
X
,
Y
]
w_{0}=[X,Y]
, we show that
φ
w
0
∗
4
\varphi _{w_{0}}^{*4}
is (FRS) for any semisimple Lie algebra, improving a result of Aizenbud-Avni, and obtaining applications in representation growth of compact
p
p
-adic and arithmetic groups.
The singularity properties we consider, such as the (FRS) property, are intimately connected to the point count of fibers over finite rings of the form
Z
/
p
k
Z
\mathbb {Z}/p^{k}\mathbb {Z}
. This allows us to relate them to properties of some natural families of random walks on finite and compact
p
p
-adic groups. We explore these connections, characterizing some of the singularity properties discussed in probabilistic terms, and provide applications to
p
p
-adic probabilistic Waring type problems.