scholarly journals Soficity, short cycles, and the Higman group

2018 ◽  
Vol 371 (4) ◽  
pp. 2771-2795
Author(s):  
Harald A. Helfgott ◽  
Kate Juschenko
Keyword(s):  
Higman Group

scholarly journals Cohomological finiteness properties of the Brin–Thompson–Higman groups 2V and 3V

2013 ◽  
Vol 56 (3) ◽  
pp. 777-804 ◽  
Author(s):  
Dessislava H. Kochloukova ◽  
Conchita Martínez-Pérez ◽  
Brita E. A. Nucinkis
Keyword(s):  
Finiteness Properties ◽  
Higman Group ◽  
Finitely Presented

AbstractWe show that Brin's generalizations 2V and 3V of the Thompson–Higman group V are of type FP∞. Our methods also give a new proof that both groups are finitely presented.

scholarly journals FACTORIZATIONS OF THE THOMPSON–HIGMAN GROUPS, AND CIRCUIT COMPLEXITY

2008 ◽  
Vol 18 (02) ◽  
pp. 285-320 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Word Problem ◽  
Word Length ◽  
Direct Limit ◽  
Circuit Complexity ◽  
Symmetric Groups ◽  
Boolean Circuits ◽  
Generating Set ◽  
Higman Group

We consider the subgroup lpGk,1 of length preserving elements of the Thompson–Higman group Gk,1 and we show that all elements of Gk,1 have a unique lpGk,1 · Fk,1 factorization. This applies to the Thompson–Higman group Tk,1 as well. We show that lpGk,1 is a "diagonal" direct limit of finite symmetric groups, and that lpTk,1 is a k∞ Prüfer group. We find an infinite generating set of lpGk,1 which is related to reversible boolean circuits. We further investigate connections between the Thompson–Higman groups, circuits, and complexity. We show that elements of Fk,1 cannot be one-way functions. We show that describing an element of Gk,1 by a generalized bijective circuit is equivalent to describing the element by a word over a certain infinite generating set of Gk,1; word length over these generators is equivalent to generalized bijective circuit size. We give some coNP-completeness results for Gk,1 (e.g., the word problem when elements are given by circuits), and [Formula: see text]-completeness results (e.g., finding the lpGk,1 · Fk,1 factorization of an element of Gk,1 given by a circuit).

CIRCUITS, THE GROUPS OF RICHARD THOMPSON, AND coNP-COMPLETENESS

2006 ◽  
Vol 16 (01) ◽  
pp. 35-90 ◽  
Author(s):  
JEAN-CAMILLE BIRGET
Keyword(s):  
Simple Group ◽  
Word Problem ◽  
Partial Transformation ◽  
Transformation Groups ◽  
Combinational Circuits ◽  
Finitely Generated ◽  
Finitely Presented Group ◽  
Thompson Groups ◽  
Higman Group ◽  
Finitely Presented

We construct a finitely presented group with coNP-complete word problem, and a finitely generated simple group with coNP-complete word problem. These groups are represented as Thompson groups, hence as partial transformation groups of strings. The proof provides a simulation of combinational circuits by elements of the Thompson–Higman group G3,1.

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