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LSpice
LSpice
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Let $F$ be a non-abelian free group and let $G=\prod_\omega F$ be the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharlampovich and Myasnikov ["Implicit function theorem over free groups and genus problem"["Implicit function theorem over free groups and genus problem", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman].

Granted, this is an example of an unreasonably bad phenomenon, not an unreasonably good phenomenon, but I still couldn't believe my ears when I was told it (by Lars Louder).

Let $F$ be a non-abelian free group and let $G=\prod_\omega F$ be the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharlampovich and Myasnikov ["Implicit function theorem over free groups and genus problem", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman].

Granted, this is an example of an unreasonably bad phenomenon, not an unreasonably good phenomenon, but I still couldn't believe my ears when I was told it (by Lars Louder).

Let $F$ be a non-abelian free group and let $G=\prod_\omega F$ be the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharlampovich and Myasnikov ["Implicit function theorem over free groups and genus problem", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman].

Granted, this is an example of an unreasonably bad phenomenon, not an unreasonably good phenomenon, but I still couldn't believe my ears when I was told it (by Lars Louder).

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HJRW
HJRW
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Let $F$ be a non-abelian free group and let $G=\prod_\omega F$, be the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharlampovich and Myasnikov ["Implicit function theorem over free groups and genus problem", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman].

Granted, this is an example of an unreasonably bad phenomenon, not an unreasonably good phenomenon, but I still couldn't believe my ears when I was told it (by Lars Louder).

Let $F$ be a non-abelian free group and let $G=\prod_\omega F$, the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharlampovich and Myasnikov ["Implicit function theorem over free groups and genus problem", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman].

Granted, this is an example of an unreasonably bad phenomenon, not an unreasonably good phenomenon, but I still couldn't believe my ears when I was told it (by Lars Louder).

Let $F$ be a non-abelian free group and let $G=\prod_\omega F$ be the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharlampovich and Myasnikov ["Implicit function theorem over free groups and genus problem", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman].

Granted, this is an example of an unreasonably bad phenomenon, not an unreasonably good phenomenon, but I still couldn't believe my ears when I was told it (by Lars Louder).

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HJRW
HJRW
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  • 75
  • 152

Let $F$ be a non-abelian free group and let $G=\prod_\omega F$, the direct product of infinitely many copies of $F$. Then the abelianisation of $G$ has torsion (of order $2$), by a theorem of Kharlampovich and Myasnikov ["Implicit function theorem over free groups and genus problem", Knots, braids, and mapping class groups—papers dedicated to Joan S. Birman].

Granted, this is an example of an unreasonably bad phenomenon, not an unreasonably good phenomenon, but I still couldn't believe my ears when I was told it (by Lars Louder).

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