In 1983 Culler and Shalen established a way to construct essential surfaces in a 3-manifold from ideal points of the $$\mathrm {SL}_2$$ SL 2 -character variety associated to the 3-manifold group. We present in this article an analogous construction of certain kinds of branched surfaces (which we call essential tribranched surfaces ) from ideal points of the $$\mathrm {SL}_n$$ SL n -character variety for a natural number n greater than or equal to 3. Further we verify that such a branched surface induces a nontrivial presentation of the 3-manifold group in terms of the fundamental group of a certain 2-dimensional complex of groups.

中文翻译：

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3-流形的高维表示和分裂的字符变体

1983 年，Culler 和 Shalen 建立了一种方法，可以从与 3-流形群相关的 $$\mathrm {SL}_2$$ SL 2 字符变体的理想点构造 3-流形中的基本曲面。我们在本文中介绍了从 $$\mathrm {SL}_n$$ SL n 字符变体的理想点对某些类型的分支表面（我们称之为基本三分支表面）的类似构造，用于自然数 n 大于或等于 3。此外，我们验证了这样的分支表面会根据某个二维群的基本群来诱导 3-流形群的非平凡表示。