Space functions of groups
Alexander Olshanskii

TL;DR
This paper studies the space complexity of the word problem in finitely presented groups, introducing space functions as a geometric analog and providing criteria for polynomial space decidability.
Contribution
It defines space functions for groups, relates them to the word problem complexity, and establishes a criterion linking polynomial space solvability to subgroup embeddings.
Findings
Space functions measure the memory needed to reduce words to identity.
A group's word problem is polynomial space decidable iff it embeds into a group with polynomial space function.
Provides a geometric perspective on computational complexity in group theory.
Abstract
We consider space functions of finitely presented groups (These functions have a natural geometric analog.) To define we start with a word over of length at most equal to 1 in and use relations from for elementary transformations to obtain the empty word; bounds from above the tape space (or computer memory) one needs to transform any word of length at most vanishing in to the empty word. One of the main obtained results is the following criterion: A finitely generated group has decidable word problem of polynomial space complexity if and only if is a subgroup of a finitely presented group with a polynomial space function.
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
Taxonomy
TopicsGeometric and Algebraic Topology · Advanced Operator Algebra Research · Finite Group Theory Research
