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added links to two papers on arxiv
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edit approved May 5, 2020 at 4:49
Harry Richman
edit approved May 5, 2020 at 4:49
Harry Richman
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There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantz, Steven, Quotient isomorphism invariants of a finitely generated Coxeter groupQuotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groupsAngle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.

There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantz, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.

There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantz, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.

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user6976
user6976

There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by CoxterCoxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, TschantzkTschantz, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.

There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantzk, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.

There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxeter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantz, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.

Source Link
user6976
user6976

There are isomorphic Coxeter groups with different Coxeter diagrams. So a simple answer to your question is "no". Nevertheless, the sets of isomorphism classes of Coxeter groups given by Coxter diagrams are not very large, and that information can be viewed as the "almost yes" answer to your question. See, for example Mihalik, Michael, Ratcliffe, John, Tschantzk, Steven, Quotient isomorphism invariants of a finitely generated Coxeter group. Aspects of infinite groups, 212–227, Algebra Discrete Math., 1, World Sci. Publ., Hackensack, NJ, 2008 or Marquis, Timothée, Mühlherr, Bernhard, Angle-deformations in Coxeter groups. Algebr. Geom. Topol. 8 (2008), no. 4, 2175–2208 and the references there.