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The Monster Lie algebra $\mathfrak{m}$, which admits an action of the Monster
finite simple group $\mathbb{M}$, was introduced by Borcherds as part of his
work on the Conway-Norton Monstrous Moonshine conjecture. Here we construct an
analog $G(\frak m)$ of a Lie group, or Kac-Moody group, associated to $\frak
m$. The group $G(\frak m)$ is given by generators and relations, analogous to
the Tits construction of a Kac-Moody group. In the absence of local nilpotence
of the adjoint representation of $\frak m$, we introduce the notion of
pro-summability of an infinite sum of operators. We use this to construct a
complete pro-unipotent group $\widehat{U}^+$ of automorphisms of a completion
$\widehat{\mathfrak{m}}=\frak n^-\ \oplus\ \frak h\ \oplus\ \widehat{\frak
n}^+$ of $\mathfrak{m}$, where $\widehat{\frak n}^+$ is the formal product of
the positive root spaces of $\frak m$. The elements of $\widehat{U}^+$ are
pro-summable infinite series with constant term 1. The group $\widehat{U}^+$
has a subgroup $\widehat{U}^+_{\text{im}}$, which is an analog of a complete
unipotent group corresponding to the positive imaginary roots of $\frak m$. We
construct analogs Exp:$\widehat{\mathfrak{n}}^+\to\widehat{U}^+$ and
Ad:$\widehat{U}^+ \to Aut(\widehat{\frak{n}}^+)$ of the classical exponential
map and adjoint representation. Although the group $G(\mathfrak m)$ is not a
group of automorphisms, it contains the analog of a unipotent subgroup $U^+$,
which conjecturally acts as automorphisms of $\widehat{\mathfrak{m}}$.

We also construct groups of automorphisms of $\mathfrak{m}$, of certain
$\mathfrak{gl}_2$ subalgebras of $\mathfrak{m}$, of the completion
$\widehat{\mathfrak{m}}$ and of similar completions of $\frak m$ that are
conjecturally identified with subgroups of~$G(\mathfrak m)$.