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- W4385245656 abstract "Consider the moduli space $X_{m,m+1}$ rank-$m$ discrete subgroups of covolume equal to one in $mathbb R^{m+1}$. There is a natural action of $text{SL}(m+1,mathbb R)$ on $X_{m,m+1}$, and it turns out that for every lattice subgroup $Gamma leqtext{SL}(m+1,mathbb R)$, each orbit $x_0.Gamma$ is dense in $X_{m,m+1}$. In this paper we compute the limiting distribution of these orbits with respect to a filtration of growing norm balls, where the norm is given by the sum of squares. The main motivation for this result comes from the work of Sargent and Shapira where they studied random walks in $X_{2,3}$. Another motivation for our work is to extend the scope of applications of the duality principle in homogeneous dynamics. The moduli space $X_{m,m+1}$ identifies as $Hbackslash text{SL}(m+1,mathbb R)$, and the duality principle recasts the above problem into the problem of establishing certain volume growth properties of growing skewed balls in $H$ and proving ergodic theorems of the left action of $H$ on $text{SL}(m+1,mathbb R)/Gamma$ along the skewed balls. Specifically, we use the duality principle as developed in the work of Gorodnik and Weiss. Our ergodic theorems are proven by applying theorems of Shah building on the linearization technique. Previously, the duality principle wasn't applied in the setting where $H$ has infinitely many non-compact connected components. Our general result is in the setting where $H$ is a certain subgroup of a minimal parabolic group of $text{SL}(m+1,mathbb R)$ such that in the Levi-component there's a lattice." @default.
- W4385245656 created "2023-07-26" @default.
- W4385245656 creator A5003325916 @default.
- W4385245656 creator A5061868395 @default.
- W4385245656 date "2023-07-22" @default.
- W4385245656 modified "2023-09-27" @default.
- W4385245656 title "Limiting distribution of dense orbits in a moduli space of rank $m$ discrete subgroups in $(m+1)$-space" @default.
- W4385245656 doi "https://doi.org/10.48550/arxiv.2307.12085" @default.
- W4385245656 hasPublicationYear "2023" @default.
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