The topology of parabolic character varieties of free groups
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- 作者：Indranil Biswas (1)
Carlos Florentino (2)
Sean Lawton (3)
Marina Logares (4) - 关键词：Free group ; Representation space ; Conjugacy class ; Strong deformation retraction ; Complex algebraic group ; 14L30 ; 20E05 ; 14P25 ; 14L17
- 刊名：Geometriae Dedicata
- 出版年：2014
- 出版时间：February 2014
- 年：2014
- 卷：168
- 期：1
- 页码：143-159
- 全文大小：239 KB
- 作者单位：Indranil Biswas (1)
Carlos Florentino (2)
Sean Lawton (3)
Marina Logares (4)
1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay, 400005, India
2. Departamento Matem谩tica, Instituto Superior T茅cnico, Av. Rovisco Pais, 1049-001, Lisbon, Portugal
3. Department of Mathematics, The University of Texas-Pan American, 1201 West University Drive, Edinburg, TX, 78539, USA
4. Instituto de Ciencias Matem谩ticas (CSIC-UAM-UC3M-UCM), Nicol谩s Cabrera 15, Campus Cantoblanco UAM, 28049, Madrid, Spain - ISSN：1572-9168
文摘
Let $G$ be a complex affine algebraic reductive group, and let $K\,\subset \, G$ be a maximal compact subgroup. Fix h $\,:=\,(h_{1}\,,\ldots \,,h_{m})\,\in \, K^{m}$ . For $n\, \ge \, 0$ , let $\mathsf X _{\mathbf{{h}},n}^{G}$ (respectively, $\mathsf X _{\mathbf{{h}},n}^{K}$ ) be the space of equivalence classes of representations of the free group on $m+n$ generators in $G$ (respectively, $K$ ) such that for each $1\le i\le m$ , the image of the $i$ -th free generator is conjugate to $h_{i}$ . These spaces are parabolic analogues of character varieties of free groups. We prove that $\mathsf X _{\mathbf{{h}},n}^{K}$ is a strong deformation retraction of $\mathsf X _{\mathbf{{h}},n}^{G}$ . In particular, $\mathsf X _{\mathbf{{h}},n}^{G}$ and $\mathsf X _{\mathbf{{h}},n}^{K}$ are homotopy equivalent. We also describe explicit examples relating $\mathsf X _{\mathbf{{h}},n}^{G}$ to relative character varieties.