Extremal entropy for products of Fuchsian representations
Abstract
In this paper, we count the number of (almost) extremally stretched closed geodesics for a pair of (non-conjugate) Fuchsian representations of a closed surface group, and show that it has subexponential growth. We then deduce that, as a discrete subgroup of PO(2, 1) × PO(2, 1), the growth indicator of the product representation vanishes on the boundary of the Benoist limit cone. We also prove that for a general Zariski dense Borel Anosov subgroup of PO(2, 1) × PO(2, 1), its growth indicator vanishes on at least one boundary component of the Benoist limit cone, but not necessarily on both. One may view our first result as a sharpening of Thurston's result that there is a unique geodesic lamination λ such that every measured lamination maximizing the ratio of lengths with respect to the two representations has support contained in λ. We hope this will be a starting point for a more general study of extremal entropy.
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BibTeX
@misc{canary2026extremal,
title = {{Extremal entropy for products of Fuchsian representations}},
author = {Richard Canary and Dongryul Kim},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.10325},
eprint = {2609.10325},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.10325},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Extremal entropy for products of Fuchsian representations
AU - Richard Canary
AU - Dongryul Kim
PY - 2026
DA - 2026/09/09
PB - arXiv
DO - 10.48550/arxiv.2609.10325
UR - https://arxiv.org/abs/2609.10325
AB - In this paper, we count the number of (almost) extremally stretched closed geodesics for a pair of (non-conjugate) Fuchsian representations of a closed surface group, and show that it has subexponential growth. We then deduce that, as a discrete subgroup of PO(2, 1) × PO(2, 1), the growth indicator of the product representation vanishes on the boundary of the Benoist limit cone. We also prove that for a general Zariski dense Borel Anosov subgroup of PO(2, 1) × PO(2, 1), its growth indicator vanishes on at least one boundary component of the Benoist limit cone, but not necessarily on both. One may view our first result as a sharpening of Thurston's result that there is a unique geodesic lamination λ such that every measured lamination maximizing the ratio of lengths with respect to the two representations has support contained in λ. We hope this will be a starting point for a more general study of extremal entropy.
ER -