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An old result of Karrass and Solitar from 1970 says that if $g$ is a nontrivial element in an amalgamated free product $G=A*_HB$, with $H$ malnormal in $G$, then the centralizer of $g$ in $G$ is either infinite cyclic or it is in a conjugate of a factor.

Is it sufficient to assume that $H$ is malnormal in $A$ or in $B$ ? Does the same result hold if we assume that $H$ is almost or weakly malnormal in $G$ ?

$H<G$ is almost malnormal if $gHg^{-1}\cap H$ is finite, for all $g\in G\setminus H$ and $H<G$ is weakly malnormal if there exist $g_1,...,g_n\in G$ such that $\bigcap_{k=1}^{n}{g_k H g_k^{-1}}$ is finite.

An old result of Karrass and Solitar from 1970 says that if $g$ is a nontrivial element in an amalgamated free product $G=A*_HB$, with $H$ malnormal in $G$, then the centralizer of $g$ in $G$ is either infinite cyclic or it is in a conjugate of a factor.

Is it sufficient to assume that $H$ is malnormal in $A$ or in $B$ ? Does the same result hold if we assume that $H$ is almost or weakly malnormal in $G$ ?

$H<G$ is almost malnormal if $gHg^{-1}\cap H$ is finite, for all $g\in G\setminus H$ and $H<G$ is weakly malnormal if there exist $g_1,...,g_n\in G$ such that $\bigcap_{k=1}^{n}{g_k H g_k^{-1}}$ is finite.

An old result of Karrass and Solitar from 1970 says that if $g$ is a nontrivial element in an amalgamated free product $G=A*_HB$, with $H$ malnormal in $G$, then the centralizer of $g$ in $G$ is either infinite cyclic or is in a conjugate of a factor.

Is it sufficient to assume that $H$ is malnormal in $A$ or in $B$ ? Does the same result hold if we assume that $H$ is almost or weakly malnormal in $G$ ?

$H<G$ is almost malnormal if $gHg^{-1}\cap H$ is finite, for all $g\in G\setminus H$ and $H<G$ is weakly malnormal if there exist $g_1,...,g_n\in G$ such that $\bigcap_{k=1}^{n}{g_k H g_k^{-1}}$ is finite.

An old result of Karrass and Solitar from 1970 says that if $g$ is a nontrivial element in an amalgamated free product $G=A*_HB$, with $H$ malnormal in $G$, then the centralizer of $g$ in $G$ is either infinite cyclic or it is in a conjugate of a factor.

Is it sufficient to assume that $H$ is malnormal in $A$ or in $B$ ? Does the same result hold if we assume that $H$ is almost or weakly malnormal in $G$ ?

$H<G$ is almost malnormal if $gHg^{-1}\cap H$ is finite, for all $g\in G\setminus H$ and $H<G$ is weakly malnormal if there exist $g_1,...,g_n\in G$ such that $\bigcap_{k=1}^{n}{g_k H g_k^{-1}}$ is finite.

An old result of Karrass and Solitar from 1970 says that if $g$ is a nontrivial element in an amalgamated free product $G=A*_HB$, with $H$ malnormal in $G$, then the centralizer of $g$ in $G$ is either infinite cyclic or is in a conjugate of a factor.

Is it sufficient to assume that $H$ is malnormal in $A$ or in $B$ ? Does the same result hold if we assume that $H$ is almost or weakly malnormal in $G$ ?

An old result of Karrass and Solitar from 1970 says that if $g$ is a nontrivial element in an amalgamated free product $G=A*_HB$, with $H$ malnormal in $G$, then the centralizer of $g$ in $G$ is either infinite cyclic or is in a conjugate of a factor.

Is it sufficient to assume that $H$ is malnormal in $A$ or in $B$ ? Does the same result hold if we assume that $H$ is almost or weakly malnormal in $G$ ?

$H<G$ is almost malnormal if $gHg^{-1}\cap H$ is finite, for all $g\in G\setminus H$ and $H<G$ is weakly malnormal if there exist $g_1,...,g_n\in G$ such that $\bigcap_{k=1}^{n}{g_k H g_k^{-1}}$ is finite.