GENERALIZATIONS OF HURTWITZ BASIS THEOREM FOR GROUP DIAGRAMS EMBEDDED IN Rn SPACE [indp_Farhaan_CP]

NATIONAL CONFERENCE ON MATHEMATICAL ADVANCES (NCMA 2025) 7– 8 April 2025, St Joseph’s University, Bengaluru, India. GENERALIZATIONS OF HURTWITZ BASIS THEOREM FOR GROUP DIAGRAMS EMBEDDED IN RnSPACE Mohammed Farhaan1 1BMSCE,Department of Computer Science Email: Mohammedfa[email protected](es) Abstract Geometric Group Theory is a field that studies symmetries in Mathematical Structures in a combinatorial method that provides an intuitive and sometimes visually appealing study. A group diagram or a cayley diagram is a graph where the vertices are the elements of the finite group and the edges are the generators of the finite group. The Heegaard genus of a closed, orientable 3-manifold M, denoted by g(M), is the smallest genus of any Heegaard splitting of M. A Heegaard splitting is a decomposition of Minto two handlebodies H1and H2 such that H1∩H2is a common boundary surface of genus g. Abstract This paper is a rough sketch of a proof of a generalization of Hurtwitz theorum that relates a group’s order to the surface genus of its group diagram or cayley graph To do this we employ the use of Euler characteristic for polyhedron V-E+F=2 which we then roughly define for higher dimensional connected complexes or n-cells, which already exists as the topological generalization of Euler Characteristic. A handlebody is like a surface with thickness. For example, a solid ball is a handlebody of genus 0, a solid torus (a thick donut) is a handlebody of genus 1, and a double torus with thickness is a handlebody of genus 2. It is the 3D version of a surface with holes. Abstract We then search for a suitable definition of higher dimensional genus so as to generalize our surface genus. We then construct our proof similar to how the hurtwitz basis was proven for 2d group diagrams (embedded in higher space depending on context) We then construct some cases to get basic info on the structure of the group by putting arbitrary values for the genus or number if generators We finish with a proposal of Research groups being launched for topics like Tropical Geometry and Complex ordered differential equations Keywords: Group Diagram, Euler Characteristic, Heegaard Genus, Geometric Group Theory 2020 AMS Subject Classification: 20F65 Introduction This paper talks about topics in the field of combinatorical Group theory or geometric group theory. A Group is a Mathematical structure observed in highly symmetrical structures and objects. They are defined as a set with a Binary operation that obeys rules like associativity, algebraic closure such that it has an identity element and an inverse for all elements. NOTE: We may also use this definition for objects with unary operations by introducing a placeholder element. Introduction For this we use the Topological definition of the Euler Characterisitc The polyhedral surfaces discussed above are, in modern language, two-dimensional finite CW-complexes. (When only triangular faces are used, they are two-dimensional finite simplicial complexes.) For a simplicial complex, the Euler characteristic equals the alternating sum χ=C0−C1+C2−C3+· · · , where Cndenotes the number of n-simplices in the complex. Introduction First we ask the question that what is the Hurtwitz Theorem and why is it important. Without the Hurtwitz Theorum the Cayley graph is only useful as a pretty picture. Through the hurtwitz Theorum and its generalizations we may extend the study from just R3 embedded, finite Cayley graphs to Rnembedded, finite Cayley graphs to perhaps infinite Group diagrams. Cayley diagrams are also super useful in recognizing group structure outside math research. You may refer the Dehn publication for the proof of Hurtwitz Theorem on the Order of Groups some of its conclusions are as follows: Some Hurtwitz results 1. If the surface genus p ≥1 and n ≥3 then: p≥1 + N 4(1) 2. if the surface genus ≥1 and n= 3 such that exactly one generator has order 3 and the rest have order 2 p≥1 + N 12 (2) Hurwitz basis for group diagram embedded in k-space dimensions Let sibe the generators of a finite group G, such that: ∃mi∈Z+∀smi i= 1 sm1 1=sm2 2=· · · =smn n= 1 V−E+F= 2 we know that V−E+F= 2 −2p Tropical Geometry Complex-Ordered Derivatives Reference 1. Max Dehn(1987),Papers in Group Theory and Topology,Springer Verlag New-york.