Block mapping class groups and their finiteness properties.

Area of Science:

• Topology and Geometric Group Theory
• Algebraic Topology
• Low-Dimensional Topology

Background:

• The study of mapping class groups is central to understanding the topology of surfaces.
• Investigating subgroups with specific algebraic properties, such as being of type F, is crucial for classifying these groups.
• The concept of "block decomposition" and "eventually acting like" provides a novel framework for constructing and analyzing subgroups.

Purpose of the Study:

• To construct and analyze a new family of subgroups within the mapping class group of modified surfaces.
• To establish a criterion for these constructed subgroups to be of type F.
• To demonstrate the existence of subgroups that exhibit specific homological finiteness properties.

Main Methods:

• Construction of a family of subgroups (denoted G_K) that preserve a block decomposition of modified surfaces.
• Analysis of the 'eventual action' of elements of G_K on the surface, relating them to a prescribed subgroup H.
• Utilizing properties of symmetric Thompson groups and homological type (F_n) to characterize the constructed subgroups.

Main Results:

• The constructed group G_K surjects onto a Farley-Hughes symmetric Thompson group, positively answering a related question.
• The main result establishes that G_K is of type F if and only if the prescribed subgroup H is of type F.
• For any genus g and any n, a subgroup is constructed that is of type F_n but not of type F_{n+1}, containing mapping class groups of compact surfaces.

Conclusions:

• The research provides a method for constructing subgroups of mapping class groups with controlled algebraic properties.
• The criterion for type F property offers a significant advancement in understanding the homological finiteness of these groups.
• The work yields explicit examples of subgroups with specific homological finiteness properties, enriching the landscape of geometric group theory.