Twisted Conjugacy in Big Mapping Class Groups

Publication date: 13 December 2021

Abstract: Let

G

be a group and

v a r p h i

be an automorphism of

G

. Two elements

x, y

of

G

are said to be

v a r p h i

-twisted conjugate if

y = g x v a r p h i (g)^{- 1}

for some

g i n G

. A group

G

has the

R_{i n f t y}

-property if the number of

v a r p h i

-twisted conjugacy classes is infinite for every automorphism

v a r p h i

of

G

. In this paper we prove that the big mapping class group

M C G (S)

possesses the

R_{i n f t y}

-property under some suitable conditions on the infinite-type surface

S

. As an application we also prove that the big mapping class group possesses the

R_{i} n f t y

-property if and only if it satisfies the

S_{i n f t y}

-property.

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