# Deepfake Video Detection Using Generative Convolutional Vision Transformer

Deressa Wodajo
Jimma University
<EMAIL_ADDRESS>Solomon Atnafu (PhD)
Addis Ababa University
<EMAIL_ADDRESS>Zahid Akhtar (PhD)
State University of New York Polytechnic Institute, USA
<EMAIL_ADDRESS>
###### Abstract

Deepfakes have raised significant concerns due to their potential to spread
false information and compromise digital media integrity. In this work, we
propose a Generative Convolutional Vision Transformer (GenConViT) for deepfake
video detection. Our model combines ConvNeXt and Swin Transformer models for
feature extraction, and it utilizes Autoencoder and Variational Autoencoder to
learn from the latent data distribution. By learning from the visual artifacts
and latent data distribution, GenConViT achieves improved performance in
detecting a wide range of deepfake videos. The model is trained and evaluated
on DFDC, FF++, DeepfakeTIMIT, and Celeb-DF (v$2$) datasets, achieving high
classification accuracy, F1 scores, and AUC values. The proposed GenConViT
model demonstrates robust performance in deepfake video detection, with an
average accuracy of 95.8% and an AUC value of 99.3% across the tested
datasets. Our proposed model addresses the challenge of generalizability in
deepfake detection by leveraging visual and latent features and providing an
effective solution for identifying a wide range of fake videos while
preserving media integrity. The code for GenConViT is available here.

Keywords: Deep Learning, Deepfakes, Deepfakes Detection, Vision Transformer,
Generative models

##  1 Introduction

Deepfakes are hyper-realistic manipulated media generated using various
advanced Deep Learning (DL) techniques. Deepfake videos are produced by
superimposing one person’s facial features onto another person’s face through
techniques such as face replacement, facial reenactment, face editing, and
complete face synthesis [46, 63], among others. Recently, deepfakes have
become a public concern due to their potential for misuse and the spread of
false information [10, 24, 72]. Various Deepfake creation methods are readily
available for use, and anyone interested can use them to easily modify
existing media, presenting a false representation of reality where individuals
appear to be speaking or performing actions that never actually took place.
Additionally, the manipulation of the videos is done on a frame-by-frame
basis, making deepfakes seem more realistic and believable. Consequently,
deepfake creation techniques have been used to manipulate images and videos,
causing celebrities and politicians to be depicted as saying or doing things
that are untrue.

The applications of Deepfake videos can range from creative uses such as
creating realistic special effects or replacing an actor in the film and
entertainment industry, to potentially harmful attacks, such as using
deepfakes to create misleading videos for criminal purposes [51, 35]. In the
realm of politics, deepfakes have been utilized to create fake political
videos, war propaganda, and videos with the intention of influencing
elections. This has led to widespread concern about the potential for
malicious actors to use deepfakes to spread false information, erode the
credibility of political candidates, or even pose a threat to national
security. The dissemination of misleading and false information through
deepfakes can have real-life consequences, making it imperative to address the
need for accurate and reliable deepfake video detection techniques.

Deepfake video detection attempts to detect if a given video has been tampered
with or manipulated. The challenge of detecting deepfakes has inspired
researchers and technology companies to develop various deep-learning methods
to identify tampered videos [65, 33, 10, 46, 43]. Currently, deepfake
detection methods often rely heavily on visual features [1, 70, 64, 47]. The
visual features-based detection can be on a frame-per-frame basis [21] or
temporal relationship between differences between frames through time [17,
28]. However, these methods mostly encounter challenges in detecting deepfakes
that differ from the training data, thus, failing sufficient detection. In
recent years, the rapid progress in advanced generative models like Generative
Adversarial Networks (GAN) [15, 27, 48], Variational Autoencoders (VAE) [31,
32], and Diffusion Models (DM) [11] has made it even more challenging to
detect deepfakes based on visual artifacts alone since the deepfakes leave
small to no trace of visual clues that help to distinguish them from the
images from the real world. Several authors have proposed alternative
detection methods, such as utilizing biological signals [8], geometric
features [69], frequency information [66], spatial features with temporal
information [73, 30] and generative adversarial network fingerprints [71], as
potential solutions to the difficulties of detecting deepfakes. Another
approach to spotting deepfake videos involves examining the consistency of
pixels or groups of similar pixels [18].

Current Deepfake video detection methods have a competitive result in
identifying manipulated videos, but they often fail to generalize in more
diverse videos, particularly in environments with different facial poses,
light angles, and movements. Based on our observation of current detection
methods, we propose a novel architecture called the Generative Convolutional
Vision Transformer (GenConViT), aimed at detecting a diverse set of fake
videos. Our proposed architecture leverages both visual artifacts and latent
data distribution of the data in its detection process. GenConViT has two main
components: the generative part and the feature extraction part.

The generative part utilizes two different models (i.e., an Autoencoder (AE)
[3] and VAE), to learn the latent data distribution of the training data. The
feature extraction component employs ConvNext [45] and the Swin Transformer
[44] to extract relevant visual features. Our proposed model addresses the
limitations of current detection methods by learning both the visual artifacts
and the latent data distribution, making it capable of detecting a wider range
of fake videos. Extensive experimental results demonstrate that GenConViT has
competitive results compared to the other models proposed for deepfake
detection.

In this paper, we propose the following contributions:

  * •

GenConViT: We propose Generative Convolution Vision Transformer for deepfake
video detection. We use AE and VAE to generate an image, extract the visual
features of the original and generated image using ConvNext and the Swin
Transformer, and then use the extracted features to detect whether the video
is a deepfake video.

  * •

By training our model on a large dataset and detecting both visual and latent
features, our proposed method aims to fill in the gap of generalizability of
deepfake detection, compared to previously proposed models.

The reminder of this article is structured as follows. An overview of the
existing works is presented in Section $2$. The proposed Generative
Convolutional Vision Transformer (GenConViT) deepfake detection framework and
datasets are detailed in Section $3$. Experiments are discussed in Section
$4$. The conclusions are drawn in Section $5$.

##  2 Related Work

In recent years, deepfakes have become more and more realistic as well as
becoming harder to detect. In this section, we will discuss the various deep
learning methods used to create and detect deepfakes.

###  2.1 Deepfake Generation

Deepfake videos are created by advanced Deep Learning (DL) methods, such as
GANs, VAE, and DMs. Deepfake was first introduced in 2017 on the social media
platform Reddit by a user called Deepfakes to showcase a DL technique he had
developed to generate Deepfake videos [51]. The videos and the accompanying
code he released garnered significant attention, and soon people started to
explore other ways to create a hyper-realistic video. Some widely used DL
techniques to create deepfakes include face synthesis, face reenactment, face
replacement, identity swapping, attribute manipulation, and expression swap,
to name a few. GANs are a type of GM that consists of two Neural Networks
(NNs), a generator $G$, which takes in random noise and produces synthetic
data, and the discriminator $D$, which determines whether the input sample is
real or fake. Another NN used to create Deepfakes is VAE, which consists of an
Encoder and a Decoder NNs. The Encoder encodes the input image into a lower
dimensional latent space, and the decoder reconstructs an image from the
latent space. Convolutional Neural Networks (CNN) [52, 19] are also used for
face synthesis by learning the mapping between a face image and its
attributes, such as facial expression, age, and gender.

Face synthesis [25] is a method of synthesizing desired non-existent face
images based on a given input image. In Face synthesis, generator $G$
generates a face image, and Discriminator $D$ discriminates whether the sample
is real or fake. Face reenactment [61, 22, 58] is a face synthesis method that
transfers source face attributes to a target face while preserving the
appearance and identity of the target face’s features. Face2Face [61] is a
real-time face reenactment technique that animates the facial expressions of
the target video by using a source actor and generates manipulated output
video. Some other examples of face synthesis methods include frontal view
synthesis, changing the facial pose from an input image, altering facial
attributes, aging the face to create diverse and realistic results. Face
swapping [36, 37, 75] is the process of replacing a person’s face in an image
or video with the face of another person to creates a non-existent realistic
face.

Several different types of GANs have been proposed for face synthesis,
including Progressive GANs [26], Wasserstein GANs [2], and Style-Based GANs
[27]. Progressive GANs allow for the generation of high-resolution images by
gradually increasing the resolution of the generated images throughout the
training process. Wasserstein GANs improve the stability of the GAN training
process by using a different loss function. Style-Based GANs allow for the
control of certain style aspects of the generated faces, such as facial
expression and hair style. Conditional GANs [48], also known as supervised
GANs, use labeled data to generate facial semantics. Paired image-to-image
translation GANs [23, 53] are a type of conditional GAN that translate an
input image from one domain to another, given input-output pairs of images as
training data. Pix2Pix is one example of a paired image-to-image translation
GAN.

###  2.2 Deepfake Detection

A key question in the deepfake detection pipeline is determining whether a
given video is fake or real. With the advancement of deep learning methods
that can create hyperrealist images, deepfake detection has increasingly
become a challenging task. To this effect, various authors have proposed
deepfake detection techniques that use different approaches, including visual
features, biological signals, and frequency information, to name a few[16].

Several deepfake detection methods rely on extracting visual features from
manipulated videos. MesoNet [1] is a deepfake detection methods that uses CNNs
to extract mesoscopic properties and identify deepfakes created with
techniques like Deepfake [49] and Face2Face [61]. Nguyen et al. [50] proposed
a model that incorporates a combination of VGG-19 and capsule networks to
learn complex hierarchical representations for detecting various types of
forgery, including FaceSwap [36], Facial Reenactment [61], replay attacks, and
AI-generated videos. Yang et al. [70] proposed a model to compare 3D head
poses estimated from all facial landmarks. The method considers that splicing
synthesized face regions into original images can introduce errors in landmark
locations. The landmark location errors can be detected by comparing the head
poses estimated from the facial landmarks. Li and Lyu [40] proposed a CNN
model for deepfake detection by identifying face-warping artifacts. Their
method leverages the limitations of deepfake algorithms that generate face
images of lower resolutions, which results in distinctive warping artifacts
when the generated images are transformed to match the original faces in the
deepfake creation pipeline. By comparing the Deepfake face region with
surrounding pixels, resolution inconsistencies caused by face warping are
identified. Li et al. [38] proposed a technique called Face X-ray that detects
the blending boundaries of images and reveals whether an input face image has
been manipulated by blending two images from different sources. Sun et al.
[59] proposed a virtual-anchor-based approach to robustly extract the facial
trajectory, capturing displacement information. They constructed a network
utilizing dual-stream spatial-temporal graph attention and a gated recurrent
unit backbone to expose manipulated videos. The proposed method achieves
competitive results on the FaceForensics++ dataset, demonstrating its
effectiveness in detecting manipulated videos.

In addition to visual features, researchers have explored the use of
biological signals for deepfake detection. Y. Li [39] proposed a model that
combines CNN and recursive neural network (LRCN) to detect deepfake videos by
tracking eye blinking with previous temporal knowledge. Chintha et al. [7]
proposed a modified XceptionNet architecture, which incorporates visual
frames, edge maps, and dense optical flow maps alongside RGB channel data to
target low-level features. The architecture isolates deepfakes at the instance
and video levels, making the technique effective in detecting deepfakes. Zhao
et al. [74] proposed a pair-wise self-consistency learning with an
inconsistency image generator to train a ConvNet [45] that extracts local
source features and measures their self-consistency to identify Deepfakes.

D. Kim and K. Kim [30] proposed a facial forensic framework that uses pixel-
level color features in the edge region of an image and a 3D-CNN
classification model to interpret the extracted color features spatially and
temporally for generalized and robust face manipulation detection. Sabir et
al. [55] proposed a Recurrent convolutional model, whereas [68] proposed a
Convolutional Vision Transformer (CViT) model, Y. Heo [21] introduced an
improved Vision Transformer model with vector-concatenated CNN feature and
patch-based positioning while [20] used Vision Transformer with distillation
and [9] combined EfficientNet and Vision Transformers (ViT) to detect
deepfakes.

##  3 The Proposed Generative Convolutional Vision Transformer (GenConViT)
Deepfake Detection Framework

In this section, we present the dataset we used, the preprocessing techniques,
and our proposed Generative Convolutional Vision Transformer (GenConViT) for
Deepfake video detection. The proposed model consists of three main steps:
$1)$ video preprocessing, $2)$ feature extraction and reconstruction, and $3)$
video classification.

###  3.1 Datasets

In our work, we utilized five datasets (i.e., DFDC [13, 12], TrustedMedia (TM)
[6], DeepfakeTIMIT (TIMIT) [34, 56], Celeb-DF (v$2$) [42, 41], and
FaceForensics++ (FF++) [54]) to train, validate, and test our model. DFDC and
FF++ are well-known benchmark datasets for deepfake detection, while TM is a
relatively new dataset that provides a diverse range of deepfake manipulation
techniques, primarily of Asian descent.

The DFDC dataset is the largest publicly available dataset and contains over
$100,000$ high resolutions of real and fake videos. The dataset is created
using $3,426$ volunteers, and the videos are captured in various natural
settings, different angles, and lighting conditions. The dataset is created
using eight deepfake creation techniques.

The FF++ dataset comprises $1,000$ original videos collected from YouTube,
which have been manipulated using four automated face manipulation methods:
Deepfakes, Face2Face, FaceSwap, and NeuralTextures [60]. The dataset includes
a $c23$ and a $c40$ compression scheme and various video resolutions. The TM
dataset consists of $4,380$ fake and $2,563$ real videos, with multiple video
and audio manipulation techniques. The TM dataset is used only in the training
phase. The Celeb-DF (v$2$) dataset consists of $890$ real videos and $5,639$
videos deepfake videos. We used all datasets to train, validate, and test our
model.

###  3.2 Video Preprocessing

The preprocessing component in Deep Learning (DL) plays a critical role in
preparing raw datasets for training, validating, and testing DL models. The
proposed model focuses on the face region, which is crucial in Deepfake
generation and synthesis mechanisms. We preprocess the videos using a series
of image processing operations. These operations include the following steps:
$(1)$ Extracting the face region from each videos using OpenCV,
face_recognition [14], and BlazeFace [4] face recognition deep learning
libraries; $(2)$ Resizing the input image to a $224\times 224$ RGB format,
where the dimensions of the input image are $H\times W\times C$ with $H=224$
representing the height, $W=224$ representing the width, and $C=3$
representing the RGB channels; and $(3)$ verifying extracted face region
images quality manually.

We extracted approximately $30$ frames per video from each dataset to ensure
diversity in our training data. To mitigate the ratio between fake and real
videos in DFDC and TM datasets, we extracted a higher number of frames from
the real videos. The DFDC dataset has a ratio of $6:1$ for fake to real
videos, and TM has $~{}2:1$ ratio. After the face regions were extracted, we
manually review them. As noted in [12], deepfake videos may contain pristine
frames within them and that the face region may not always be accurately
detected by the deep learning frameworks used to extract them. To address this
issue, we manually review the images and exclude images that did not contain a
face or were deemed to be real image within the fake class, resulting in a
total of $1,004,810$ images for training. This approach allowed us to curate a
fake class dataset comprising only relevant and potentially manipulated face
images. To train, validate, and test our model, the images were divided into a
ratio of $80:15:5$, respectively. To evaluate our model’s performance on
videos, we held out $3,972$ videos from both DFDC, DeepfakeTIMIT, Celeb-DF
(v$2$) , and FF++ datasets for testing. All the facial images are saved to a
$224\times 224$ RGB format.

###  3.3 Generative Convolutional Vision Transformer (GenConViT)

The Generative Convolutional Vision Transformer (GenConViT) model generates
latent spaces of video frames and extracts visual clues and hidden patterns
within them to determine if a video is real or fake. The proposed GenConViT
model, as shown in Fig. 1, has two independently trained networks and four
main modules: an Autoencoder (AE), a Variational Autoencoder (VAE), a ConvNeXt
layer, and a Swin Transformer. The first network ($A$ part in Fig. 1) includes
an AE, a ConvNeXt layer, and a Swin Transformer, while the second network ($B$
part in Fig. 1) includes a VAE, a ConvNeXt layer, and a Swin Transformer. The
first network uses an AE to generate Latent Feature (LF) space of an input
images to maximize the class prediction probability, indicating the likelihood
that a given input is deepfake. The second network uses a VAE to reconstruct
images to maximize class prediction probability and minimize the loss distance
between the sample image and the reconstructed image. Both AE and VAE models
extract LFs from input video frames, which capture hidden patterns and
correlations present in the learned deepfake visual artifacts. The ConvNeXt
and Swin transformer models form a hybrid model ConvNeXt-Swin. The ConvNeXt
model acts as the backbone of the hybrid model, using a CNN to extract
features from the input frames. The Swin Transformer, with its hierarchical
feature representation and attention mechanism, further extracts the global
and local features of the input images. The GenConViT two networks have each
two ConvNeXt-Swin models, and both take in a $224\times 224$ RGB image and an
LF of AE ($I_{A}$) or VAE ($I_{B}$). The use of the ConvNeXt-Swin hybrid model
enables the learning of relationships among the extracted LFs by the AE and
VAE.

Figure 1: The Proposed Generative Convolutional Vision Transformer (GenConViT)
Deepfake Detection Framework.

####  3.3.1 Autoencoder and Variational Autoencoder

An AE and a VAE are NNs that consist of two networks: an Encoder and a
Decoder. The Encoder of AE maps an input image $X\in\mathbb{R}^{H\times
W\times C}$ to a latent space $Z\in\mathbb{R}^{H^{\prime}\times
W^{\prime}\times{\textnormal{K}}}$, where ${K}$ is the number of channels
(features) in the output and ${H^{\prime}}$ and ${W^{\prime}}$ are the height
and width of the output feature map, respectively, and its decoder maps the
latent space $Z\in\mathbb{R}^{H^{\prime}\times
W^{\prime}\times{\textnormal{K}}}$ to an output image
$X^{\prime}\in\mathbb{R}^{H\times W\times C}$. The Encoder of VAE maps an
input image $X$ to a probability distribution over a latent space
$Z^{\prime}\in\mathbb{R}^{K},Z^{\prime}\sim\mathcal{N}(\mu,\sigma^{2})$, and
$\mu$ and $\sigma^{2}$ are the mean and variance of the learned distribution,
respectively. The Encoder of the AE is composed of five convolutional layers
with width starting from $3$ up to $256$, with kernels of size $3\times 3$ and
a stride of $2$. Each convolutional layer is followed by ReLU non-linearity
and Max-pooling of kernel size $2\times 2$ and stride $2$. The output of the
Encoder is a $256\times 7\times 7$ down-sampled LF. The Decoder is composed of
five transposed convolutional layers with width starting from $256$ up to $3$,
with kernels of size $2\times 2$ and stride of $2$. Each transposed
convolutional layer is followed by ReLU non-linearity. The output of the
Decoder, $I_{A}$, is a reconstructed feature space of the input image with
dimensions $H\times W\times C$. In this case, $I_{A}$ has dimensions
$224\times 224\times 3$. The detailed configuration is shown in Table $1$.

The goal of VAE is to learn a meaningful latent representation of the input
image and reconstruct the input image by performing random sampling of the
latent space while minimizing the reconstruction loss. The Encoder of the VAE
is composed of four convolutional layers with width starting from $3$ up to
$128$, with kernels of size $3\times 3$ and a stride of $2$. Each
convolutional layer is followed by Batch Normalization (BN) and LeakyReLU non-
linearity. The output of the Encoder is a $1$-dimensional vector of length
$12544$ representing the latent distributions. The Decoder is composed of four
transposed convolutional layers with width starting from $256$ up to $3$, with
kernels of size $2\times 2$ and stride of $2$. Each transposed convolutional
layer is followed by LeakyReLU non-linearity. The output of the Decoder,
$I_{B}$, is a reconstructed feature space of the input image with dimensions
$H/2\times W/2\times C$. In this case, $I_{B}$ has dimensions $112\times
112\times 3$. The detailed configuration is shown in Table $2$. The choice of
the convolutional layers for both the AE and VAE is due to the compute power
and memory we had, model accuracy, extensive experiment, and training time
during the training of our model.

Table 1: GenConViT model Autoencoder configuration.

Network | AE Configuration
---|---
Encoder |  | Conv- | Kernel | Stride
| $3$-$16$ | $16$-$32$ | $32$-$64$ | $64$-$128$ | $128$-$256$ | $3$ | $1$
Decoder |  | ConvTranspose- |
| $256$-$128$ | $128$-$64$ | $64$-$32$ | $32$-$16$ | $16$-$3$ | $2$ | $1$

Table 2: GenConViT model Variational Autoencoder configuration.

Network | VAE Configuration
---|---
Encoder |  | Conv- | Kernel | Stride
| $3$-$16$ | $16$-$32$ | $32$-$64$ | $64$-$128$ | $3$ | $2$
Decoder |  | ConvTranspose- |
| $256$-$64$ | $64$-$32$ | $32$-$16$ | $16$-$3$ | $2$ | $2$

####  3.3.2 ConvNeXt-Swin Hybrid

The ConvNeXt-Swin Transformer architecture is a hybrid CNN-Transformer model
that combines the strengths of ConvNeXt and Swin Transformer architectures for
deepfake detection task. The ConvNeXt model is a CNN architecture that has
shown impressive performance in image recognition tasks, by extracting high-
level features from images through a series of convolutional layers. The Swin
Transformer is a transformer-based model that uses a self-attention mechanism
to extract local and global features.

The GenConViT model leverages the strengths of both architectures by using
ConvNeXt as the backbone for feature extraction and the Swin Transformer for
feature processing. In our proposed method, the ConvNeXt architecture extracts
high-level features from images, which are then passed through a HybridEmbed
module to embed the features into a compact and informative vector. The
resulting vector is then passed to the Swin Transformer model. The ConvNeXt
backbone consists of multiple convolutional layers that extract high-level
features from input images and the LFs from AE or VAE. We use pre-trained
ConvNeXt and Swin Transformer models, which are trained on an ImageNet
dataset.

After extracting learnable features by the ConvNeXt backbone, we pass the
feature maps through a HybridEmbed module. The HybridEmbed module is designed
to extract feature maps from the ConvNeXt, flatten them, and project them to
an embedding dimension of $768$. It consists of a $1\times 1$ convolutional
layer, which takes the feature maps from the backbone and reduces their
channel dimension to the desired embedding dimension. The resulting feature
maps are flattened and transposed to obtain a sequence of feature vectors,
which are then further processed by the Swin Transformer.

The GenConViT’s network $A$ consists of two Hybrid ConvNeXt-Swin models that
take in a LF of size $224\times 224\times 3$ generated by the AE ($I_{A}$) and
an input image of the same size. The models output a feature space of size
$1,000$, which is then concatenated. Since the model has only two classes
(real or fake), a linear mapping layer of size $2$ is used. Network $B$ has
the same configuration as $A$, but it uses VAE and outputs both class
prediction probability and the reconstructed image of size $112\times
112\times 3$.

##  4 Experiments

We conducted extensive experiments on various configurations of AE and VAE, as
well as different variants of CNN and Transformer models. Our findings suggest
that a hybrid architecture using ConvNeXt and Swin Transformer performs well.
Due to our limited resources and large training dataset, we implemented the
”tiny” model versions of both architectures.

###  4.1 Implementation Details

The network $A$ and $B$ were trained to classify real and fake videos while
$B$ was also trained to the reconstruct images, as shown in Fig. 2.

(a) Input Sample Images

(b) Reconstructed Images by Network $B$

Figure 2: Generated Images ($I_{B}$) from Input Samples (a) using Network $B$
(b)

Therefore, Network $A$ is trained using Cross Entropy loss while Network $B$
is trained using Cross Entropy and MSE loss. We used timm [67] library to load
the class definitions and the weight of the pretrained ConvNext and Swin
Transformer. We used two model variants, namely convnext_tiny and
swin_tiny_patch4_window7_224, both trained on ImageNet-$1$k.

Both network, $A$ and $B$, were trained using Adam optimizer with a learning
rate of $0.0001$ and weight decay of $0.0001$. The Albumentation [5] library
was used for data augmentation, and the following augmentation techniques were
used with a strong augmentation rate of $90\%$: RandomRotate, Transpose,
HorizontalFlip, VerticalFlip, GaussNoise, ShiftScaleRotate, CLAHE, Sharpen,
IAAEmboss, RandomBrightnessContrast, and HueSaturationValue. The training data
was normalized. The batch size for network $A$ is set to $32$ and for network
$B$, it was set to $16$. Both networks were trained on the DFDC, FF++, and TM
datasets for $30$ epochs. The training size of our deepfake data is as
follows: train: $826,756$, valid: $130,948$, and test: $47,106$.

###  4.2 Experimental Results and Discussion

In this section, we present the experimental results and discuss the
performance of our proposed GenConViT model.

To assess GenConViT’s performance, we used multiple evaluation metrics,
including classification accuracy, F$1$ score, Receiver Operating
Characteristic (ROC) curve, and Area Under the Curve (AUC) value. We evaluated
the model using four different datasets: DFDC, FF++, DeepfakeTIMIT, and Celeb-
DF (v$2$). The model was trained with $1,004,810$ images. Ninety percent of
the images were augmented online. This resulted in a total of $1,909,139$
images. The model was tested on $3,972$ videos across these datasets. To
predict our model’s performance, we extracted $15$ frames from each video. We
averaged the result of both networks for the final prediction. The results
demonstrate that GenConViT delivers strong performance across various
datasets. Table $3$ shows our model’s accuracy on the training, validation,
and testing dataset.

Table 3: GenConViT model accuracy on the training, validation, and testing
dataset

Dataset | GenConViT($A$) | GenConViT($B$) | GenConViT
---|---|---|---
Train | $99\%$ | $99\%$ | -
Validation | $97\%$ | $97\%$ | -
Test | $99.99\%$ | $99.99\%$ | $99.99\%$

We evaluated GenConViT classification accuracy for each of the dataset. Table
$4$ summarizes their results.

Table 4: GenConViT classification accuracy for each dataset. Dataset |  Accuracy (%)
---|---
DFDC |  $98.5$
TIMIT |  $98.28$
FF++ |  $97$
Celeb-DF (v$2$) |  $90.94$

We evaluated GenConViT classification accuracy for each Real and False classes
within each dataset. Table $5$ provides the breakdown of these results.

Table 5: GenConViT classification accuracy for each Real and Fake class in each dataset. Dataset |  REAL (%) |  FAKE (%)
---|---|---
DFDC |  $98.7$ |  $98.45$
FF++ |  $95.58$ |  $98.5$
TIMIT |  - |  $98.28$
Celeb-DF (v$2$) |  $83$ |  $98.8$

We evaluated the GenConViT performance by testing the classification accuracy
for each Real and Fake class of the two variants of the network,
GenConViT($A$) and GenConViT($B$) within each dataset. The results are
summarized on Table $6$.

Table 6: GenConViT classification accuracy for each Real and Fake class of the two Network’s $A$ and $B$ in each dataset. Dataset |  GenConViT($A$) |  GenConViT($B$)
---|---|---
DFDC |  $97.5\%$ |  $98.45\%$
FF++ |  $95.57\%$ |  $96.8\%$
TIMIT |  $97.65\%$ |  $97.81\%$
Celeb-DF (v$2$) |  $85.42\%$ |  $83.97\%$

Table $7$ summarizes the classification accuracy for each Real and Fake
classes of the two networks’: GenConViT($A$) and GenConViT($B$) within each
dataset:

Table 7: GenConViT classification accuracy for Real and Fake classes in each dataset. Dataset | GenConViT($A$) | GenConViT($B$)
---|---|---
REAL | FAKE | REAL | FAKE
DFDC | $98.7\%$ | $97.2\%$ | $98.7\%$ | $95.52\%$
FF++ | $94.12\%$ | $95.56\%$ | $95.58\%$ | $98.02\%$
TIMIT |  - | $97.5\%$ |  - | $97.8\%$
Celeb-DF (v$2$) | $70.22\%$ | $93.38\%$ | $55\%$ | $99.38\%$

We compared our results with other state-of the-art results on DFDC dataset,
as summarized in Table $8$. Previous approaches achieved accuracy values
ranging from $91.5\%$ to $98.24\%$. Notable models included Khan [28]
($91.69\%$), and Thing [62] ($92.02\%$), Selim [57] ($97.2\%$). Some models
excelled in additional metrics, such as the STDT [73] model with $97.44\%$
accuracy, $99.1\%$ AUC, and $98.48\%$ F$1$-score. In comparison to previous
approaches, our proposed model demonstrated excellent performance, achieving
an accuracy of $98.5\%$, an AUC of $99.9\%$, and an F$1$-score of $99.1\%$.

Table 8: GenConViT model accuracy and AUC on the DFDC dataset compared to with other state-of-the-art models. Model | Accuracy | AUC |  F$1$-Score
---|---|---|---
Image+Video Fusion [28] | $91.69\%$ | - | -
Selim EfficientNet B7 [57] | $97.20\%$ | $90.60\%$ | -
CViT [68] | $91.50\%$ | $91\%$ | -
Thing [62] | $92.02\%$ | $97.61\%$ | -
Random cut-out [29] | $98.24\%$ | - | -
STDT [73] | $97.44\%$ | $99.10\%$ | 98.48%
ViT with distillation [20] | - | $97.80\%$ | $91.90\%$
Heo et. al [21] | - | $97.80\%$ | $91.90\%$
Coccomini et. al [9] | - | $95.10\%$ | $88.00\%$
Ours | 98.50% | 99.90% |  99.10%

Table $9$ summarizes our model’s accuracy and AUC on the FF++ dataset compared
to with other state-of-the-art models.

Table 9: GenConViT model accuracy and AUC on the FF++ dataset compared to with other state-of-the-art models. Model |  Accuracy |  AUC |  F1-Score
---|---|---|---
Li et. al [38] |  $97.73\%$ |  $98.52\%$ |  -
Image+Video Fusion [28] |  $99.52\%$ |  $99.64\%$ |  99.28
Random cut-out [29] |  $97.00\%$ |  $99.28\%$ |  -
Ours |  $97\%$ |  99.60% |  $97.1$

Table $10$ summarizes our model’s accuracy on the three subsets of FF++
dataset compared to with other models.

Table 10: GenConViT model accuracy on the three subsets of FF++ dataset compared to with other state-of-the-art models. Model |  Deepfakes |  Face2Face |  NeuralTextures
---|---|---|---
Li et. al [38] |  $99.17\%$ |  $97.73\%$ |  -
Coccomini [9] |  $87\%$ |  - |  $69\%$
Random cut-out [29] |  $98.57\%$ |  $98.57\%$ |  $90.71\%$
Ours |  $92.27\%$ |  98% |  97%

Table $11$ presents the AUC values for each dataset, showcasing the model’s
ability to discriminate between Real and Fake classes. To further investigate
GenConViT performance, we also examined the ROC curve. Figure 3 presents the
resulting ROC curve.

Table 11: GenConViT AUC values for each dataset. Dataset |  AUC (%)
---|---
DFDC | $99.9$
FF++ | $99.6$
Celeb-DF (v$2$) | $98.1$
Figure 3: ROC curve illustrating the model’s discrimination ability between
real and fake classes.

The AUC and F$1$ score of GenConViT($A$) and GenConViT($B$) are shown in Table
$12$, which provides a summary of the results, allowing for a comparison
between the two networks. Both networks have relatively similar results, with
GenConViT($B$) is slightly outperforming GenConViT($A$).

Table 12: AUC values for GenConViT($A$) and GenConViT($B$) in each dataset. Dataset |  GenConViT($A$) |  GenConViT($B$)
---|---|---
DFDC | $99.9$ | $99.9$
FF++ | $99.1$ | $99.6$
Celeb-DF (v$2$) | $97.8$ | $94.1$
Table 13: F$1$ scores for GenConViT($A$) and GenConViT($B$) in each dataset. Dataset |  GenConViT($A$) |  GenConViT($B$)
---|---|---
DFDC | $0.984$ | $0.984$
FF++ | $0.949$ | $0.968$
Celeb-DF (v$2$) | $0.952$ | $0.890$

Additionally, our model achieved F$1$ scores of $99.1\%$, $97.1\%$, $95.5\%$,
and $91.6\%$ for the DFDC, FF++, DeepfakeTIMIT, and Celeb-DF (v$2$) datasets,
respectively. Overall, the proposed GenConViT model has an average accuracy of
$95.8\%$ and an AUC value of $99.3\%$ across the tested datasets. These
results highlight our model’s robust performance in detecting deepfake videos,
demonstrating its potential for practical applications in the field.

##  5 Conclusion

In this work, we proposed a Generative Convolutional Vision Transformer
(GenConViT) that extracts visual artifacts and latent data distributions to
detect deepfake videos. GenConViT combines ConvNext and the Swin transformer
models to learn from local and global image features of a video, as well as AE
and VAE to learn from internal data representation. Our approach aims to fill
the gap in the generalizability of deepfake detection and provides an
effective solution for identifying a wide range of fake videos while
preserving media integrity. Through extensive experiments on a diverse
dataset, including DFDC, FF++, DeepfakeTIMIT, and Celeb-DF (v$2$), our
GenConViT model demonstrated an improved and robust performance with high
classification accuracy, F$1$ scores, and AUC values. Overall, our proposed
GenConViT model offers a promising approach for accurate and reliable deepfake
video detection.

## Acknowledgement

This research was funded by Addis Ababa University Research Grant for the
Adaptive Problem-Solving Research. Reference number RD/PY-183/2021. Grant
number AR/048/2021.

## References

  * [1] D. Afchar, V. Nozick, J. Yamagishi, and I. Echizen.  MesoNet: a Compact Facial Video Forgery Detection Network.  in 2018 IEEE International Workshop on Information Forensics and Security (WIFS, 2018.
  * [2] M. Arjovsky, S. Chintala, and L. Bottou.  Wasserstein gan.  2017\.
  * [3] D. Bank, N. Koenigstein, and R. Giryes.  Autoencoders.  arXiv, 2021.
  * [4] V. Bazarevsky, Y. Kartynnik, A. Vakunov, K. Raveendran, and M. Grundmann.  Blazeface: Sub-millisecond neural face detection on mobile gpus.  2019\.
  * [5] A. Buslaev, V. I. Iglovikov, E. Khvedchenya, A. Parinov, M. Druzhinin, and A. A. Kalinin.  Albumentations: Fast and flexible image augmentations.  Information, 11, February 2020.
  * [6] W. Chen, S. L. B. Chua, S. Winkler, and S.-K. Ng.  Trusted media challenge dataset and user study.  In Proceedings of the 31st ACM International Conference on Information and Knowledge Management, 2022.
  * [7] A. Chintha, A. Rao, S. Sohrawardi, K. Bhatt, M. Wright, and R. Ptucha.  Leveraging edges and optical flow on faces for deepfake detection.  In 2020 IEEE International Joint Conference on Biometrics (IJCB). Houston, 2020.
  * [8] U. A. Ciftci and I. Demir.  Fakecatcher: Detection of synthetic portrait videos using biological signals.  2020\.
  * [9] D. Coccomini, N. Messina, C. Gennaro, and F. Falchi.  Combining efficientnet and vision transformers for video deepfake detection.  1323:219–229, 2022.
  * [10] D. Dagar and D. K. Vishwakarma.  A literature review and perspectives in deepfakes: generation, detection, and applications.  International Journal of Multimedia Information Retrieval, 11:219–289, September 2022.
  * [11] P. Dhariwal and A. Nichol.  Diffusion models beat gans on image synthesis, arxiv.  2021\.
  * [12] B. Dolhansky, J. Bitton, B. Pflaum, J. Lu, R. Howes, M. Wang, and C. C. Ferrer.  The deepfake detection challenge (dfdc) dataset, arxiv.  2020\.
  * [13] B. Dolhansky, R. Howes, B. Pflaum, N. Baram, and C. C. Ferrer.  The deepfake detection challenge (dfdc) preview dataset, arxiv.  2019\.
  * [14] A. Geitgey.  The world’s simplest facial recognition api for Python and the command line.  Github.
  * [15] I. J. Goodfellow, J. Pouget-Abadie, M. Mirza, B. Xu, D. Warde-Farley, S. Ozair, A. Courville, and Y. Bengio.  Generative adversarial networks, arxiv.  2014\.
  * [16] Diego Gragnaniello, Francesco Marra, and Luisa Verdoliva.  Detection of AI-Generated Synthetic Faces, pages 191–212.  Springer International Publishing, Cham, 2022.
  * [17] Z. Gu, Y. Chen, T. Yao, S. Ding, J. Li, F. Huang, and L. Ma.  Spatiotemporal inconsistency learning for deepfake video detection, arxiv.  2021\.
  * [18] F. Guillaro, D. Cozzolino, A. Sud, N. Dufour, and L. Verdoliva.  Trufor: Leveraging all-round clues for trustworthy image forgery detection and localization, arxiv.  2022\.
  * [19] R. Haridas and R. L. Jyothi.  Convolutional neural networks: A comprehensive survey.  International Journal of Applied Engineering Research, 14, February 2019.
  * [20] Y.-J. Heo, Y.-J. Choi, Y.-W. Lee, and B.-G. Kim.  Deepfake detection scheme based on vision transformer and distillation, arxiv.  2021\.
  * [21] Y.-J. Heo, W.-H. Yeo, and B.-G. Kim.  Deepfake detection algorithm based on improved vision transformer.  Applied Intelligence, 53:7512–7527, April 2023.
  * [22] G.-S. Hsu, C.-H. Tsai, and H.-Y. Wu.  Dual-generator face reenactment.  In 2022 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR, 2022.
  * [23] P. Isola, J.-Y. Zhu, T. Zhou, and A. A. Efros.  Image-to-image translation with conditional adversarial networks.  2018\.
  * [24] F. Juefei-Xu, R. Wang, Y. Huang, Q. Guo, L. Ma, and Y. Liu.  Countering malicious deepfakes: Survey, battleground, and horizon.  International Journal of Computer Vision, 130:1678–1734, July 2022\.
  * [25] A. Kammoun, R. Slama, H. Tabia, T. Ouni, and M. Abid.  Generative adversarial networks for face generation: A survey.  ACM Comput. Surv., 55, December 2022.
  * [26] T. Karras, T. Aila, S. Laine, and J. Lehtinen.  Progressive growing of gans for improved quality, stability, and variation.  2018\.
  * [27] T. Karras, S. Laine, and A T. Aila.  A style-based generator architecture for generative adversarial networks.  2019\.
  * [28] S. A. Khan and H. Dai.  Video transformer for deepfake detection with incremental learning.  In Virtual Event China, editor, Proceedings of the 29th ACM International Conference on Multimedia, 2021.
  * [29] S. A. Khan and D.-T. Dang-Nguyen.  Hybrid transformer network for deepfake detection, arxiv.  2022\.
  * [30] D.-K. Kim and K. Kim.  Generalized facial manipulation detection with edge region feature extraction.  In 2022 IEEE/CVF Winter Conference on Applications of Computer Vision (WACV). Waikoloa, 2022.
  * [31] D. P. Kingma and M. Welling.  Auto-encoding variational Bayes.  2022\.
  * [32] D. P. Kingma and M. Welling.  An introduction to variational autoencoders.  Foundations and Trends® in Machine Learning, 12:307–392, 2019\.
  * [33] S. Kingra, N. Aggarwal, and N. Kaur.  Emergence of deepfakes and video tampering detection approaches: A survey.  Multimedia Tools and Applications, 82:10165–10209, March 2023.
  * [34] P. Korshunov and S. Marcel.  DeepFakes: a New Threat to Face Recognition? Assessment and Detection, 2018.
  * [35] Pavel Korshunov and Sébastien Marcel.  The Threat of Deepfakes to Computer and Human Visions, pages 97–115.  Springer International Publishing, Cham, 2022.
  * [36] I. Korshunova, W. Shi, J. Dambre, and L. Theis.  Fast face-swap using convolutional neural networks.  2017\.
  * [37] L. Li, J. Bao, H. Yang, D. Chen, and F. Wen.  Faceshifter: Towards high fidelity and occlusion aware face swapping.  2020\.
  * [38] L. Li, J. Bao, T. Zhang, H. Yang, D. Chen, F. Wen, and B. Guo.  Face x-ray for more general face forgery detection, arxiv.  2020\.
  * [39] Y. Li, M.-C. Chang, and S. Lyu.  In Ictu Oculi: Exposing AI Created Fake Videos by Detecting Eye Blinking.  in 2018 IEEE International Workshop on Information Forensics and Security (WIFS), Hong Kong, Hong Kong, 2018.
  * [40] Y. Li and S. Lyu.  Exposing deepfake videos by detecting face warping artifacts.
  * [41] Yuezun Li, Pu Sun, Honggang Qi, and Siwei Lyu.  Toward the Creation and Obstruction of DeepFakes, pages 71–96.  Springer International Publishing, Cham, 2022.
  * [42] Y. Li, X. Yang, P. Sun, H. Qi, and S. Lyu.  Celeb-df: A large-scale challenging dataset for deepfake forensics.  In 2020 IEEE/CVF Conference on Computer Vision and Pattern Recognition (CVPR). Seattle, 2020.
  * [43] P. Liu, Y. Lin, Y. He, Y. Wei, L. Zhen, J. T. Zhou, R. S. M. Goh, and J. Liu.  Automated deepfake detection, arxiv.  2021\.
  * [44] Z. Liu, Y. Lin, Y. Cao, H. Hu, Y. Wei, Z. Zhang, S. Lin, and B. Guo.  Swin transformer: Hierarchical vision transformer using shifted windows, arxiv.  2021\.
  * [45] Z. Liu, H. Mao, C.-Y. Wu, C. Feichtenhofer, T. Darrell, and A S. Xie.  Convnet for the 2020s, arxiv.  2022\.
  * [46] M. Masood, M. Nawaz, K. M. Malik, A. Javed, A. Irtaza, and H. Malik.  Deepfakes generation and detection: state-of-the-art, open challenges, countermeasures, and way forward.  Applied Intelligence, 53:3974–4026, February 2023.
  * [47] F. Matern, C. Riess, and M. Stamminger.  Exploiting Visual Artifacts to Expose Deepfakes and Face Manipulations.  in 2019 IEEE Winter Applications of Computer Vision Workshops (WACVW), Waikoloa Village, HI, USA, 2019.
  * [48] M. Mirza and S. Osindero.  Conditional generative adversarial nets, arxiv.  2014\.
  * [49] H. H. Nguyen, N.-D. T. Tieu, H.-Q. Nguyen-Son, V. Nozick, J. Yamagishi, and I. Echizen.  Modular convolutional neural network for discriminating between computer-generated images and photographic images.  In 13th International Conference on Availability, Hamburg, 2018\. Reliability and Security (ARES 2018.
  * [50] H. H. Nguyen, J. Yamagishi, and I. Echizen.  Capsule-forensics: Using capsule networks to detect forged images and videos, arxiv.  2018\.
  * [51] T. T. Nguyen, Q. V. H. Nguyen, D. T. Nguyen, D. T. Nguyen, T. Huynh-The, S. Nahavandi, T. T. Nguyen, Q.-V. Pham, and C. M. Nguyen.  Deep learning for deepfakes creation and detection: A survey.  Computer Vision and Image Understanding, 223, October 2022.
  * [52] K. O’Shea and R. Nash.  An introduction to convolutional neural networks, arxiv.  2015\.
  * [53] Y. Pang, J. Lin, T. Qin, and Z. Chen.  Image-to-image translation: Methods and applications.  2021\.
  * [54] A. Rössler, D. Cozzolino, L. Verdoliva, C. Riess, J. Thies, and M. Nießner.  Faceforensics++: Learning to detect manipulated facial images, arxiv.  2019\.
  * [55] E. Sabir, J. Cheng, A. Jaiswal, W. AbdAlmageed, I. Masi, and P. Natarajan.  Recurrent convolutional strategies for face manipulation detection in videos.
  * [56] C. Sanderson and B. C. Lovell.  Multi-Region Probabilistic Histograms for Robust and Scalable Identity Inference.  in Advances in Biometrics, Berlin, 2009.
  * [57] S. Seferbekov.  A prize winning solution for dfdc challenge.  0, 6, June.
  * [58] J. Sun, Q. Deng, Q. Li, M. Sun, M. Ren, and Z. Sun.  Anyface: Free-style text-to-face synthesis and manipulation.  2022\.
  * [59] Y. Sun, Z. Zhang, I. Echizen, H. H. Nguyen, C. Qiu, and L. Sun.  Face forgery detection based on facial region displacement trajectory series.  In Proceedings of the IEEE/CVF Winter Conference on Applications of Computer Vision (WACV) Workshops, 2023.
  * [60] J. Thies, M. Zollhöfer, and M. Nießner.  Deferred neural rendering: Image synthesis using neural textures.  2019\.
  * [61] J. Thies, M. Zollhöfer, M. Stamminger, C. Theobalt, and M. Nießner.  Face2face: Real-time face capture and reenactment of rgb videos.
  * [62] V. L. L. Thing.  Deepfake detection with deep learning: Convolutional neural networks versus transformers.  2023\.
  * [63] Ruben Tolosana, Ruben Vera-Rodriguez, Julian Fierrez, Aythami Morales, and Javier Ortega-Garcia.  An Introduction to Digital Face Manipulation.  In Christian Rathgeb, Ruben Tolosana, Ruben Vera-Rodriguez, and Christoph Busch, editors, Handbook of Digital Face Manipulation and Detection: From DeepFakes to Morphing Attacks, pages 3–26. Springer International Publishing, Cham, 2022.
  * [64] A. Trabelsi, M. M. Pic, and J.-L. Dugelay.  Improving deepfake detection by mixing top solutions of the dfdc.  in, 2022:30, 2022.
  * [65] P. N. Vasist and S. Krishnan.  Engaging with deepfakes: a meta-synthesis from the perspective of social shaping of technology theory.  Internet Research, 2022.
  * [66] V. Wesselkamp, K. Rieck, D. Arp, and E. Quiring.  Misleading deep-fake detection with gan fingerprints, arxiv.  2022\.
  * [67] R. Wightman and GitHub PyTorch Image Models.  2019\.
  * [68] D. Wodajo and S. Atnafu.  Deepfake video detection using convolutional vision transformer, arxiv.  2021\.
  * [69] Z. Yan, P. Sun, Y. Lang, S. Du, S. Zhang, W. Wang, and L. Liu.  Multimodal graph learning for deepfake detection, arxiv.  2023\.
  * [70] X. Yang, Y. Li, and S. Lyu.  Exposing deep fakes using inconsistent head poses.  In Icassp Ieee, editor, International Conference on Acoustics, Speech and Signal, pages 2019–2019. Processing (ICASSP, 2019.
  * [71] N. Yu, V. Skripniuk, S. Abdelnabi, and M. Fritz.  Artificial fingerprinting for generative models: Rooting deepfake attribution in training data.  2022\.
  * [72] G. P. Zachary.  Digital manipulation and the future of electoral democracy in the u.s.  IEEE Transactions on Technology and Society, 1:104–112, June 2020\.
  * [73] D. Zhang, F. Lin, Y. Hua, P. Wang, D. Zeng, and S. Ge.  Deepfake video detection with spatiotemporal dropout transformer, arxiv.  2022\.
  * [74] T. Zhao, X. Xu, M. Xu, H. Ding, Y. Xiong, and W. Xia.  Learning self-consistency for deepfake detection.  In 2021 IEEE/CVF International Conference on Computer Vision (ICCV). Montreal, 2021.
  * [75] Y. Zhu, Q. Li, J. Wang, C. Xu, and Z. Sun.  One shot face swapping on megapixels.  2022\.


We give a sharp lower bound on the size of nonabelian quotients of the surface braid group $B_n(\Sigma_g)$ and classify all quotients that attain the lower bound: Depending on $n$ and $g$, a quotient of minimum order is either a symmetric group or a 2-step nilpotent $p$-group.

§ INTRODUCTION

The Artin braid group $B_n$ arises as the fundamental group of $\UConf_n(\DD)$, the configuration space of $n$ distinct unordered points on the open disk $\DD$. One can generalize this construction to define for an oriented, closed genus $g$ surface $\Sigma_g$ the surface braid groups

$$B_n(\Sigma_g) = \pi_1(\UConf_n(\Sigma_g)).$$

It was shown by Kolay <cit.> that for $n = 3$ or $n \geq 5$, the smallest noncyclic finite quotient of $B_n$ is the symmetric group $S_n$, in the sense that $S_n$ has minimum order amongst noncyclic quotients of $B_n$ and $S_n$ is the unique noncyclic quotient of $B_n$ of minimum order.

In this paper we consider the analogous question for surface braid groups. With our main result we show that whilst $S_n$ is a quotient of $B_n(\Sigma_g)$, it is not generally the smallest nonabelian quotient, in contrast to the regular braid groups. The new minimal nonabelian quotients that arise are 2-step nilpotent $p$-groups which will be defined in <ref>.

For $g \geq 1$ there is an embedding $B_n \hookrightarrow B_n(\Sigma_g)$ <cit.>. By a braid-reduced quotient of $B_n(\Sigma_g)$ we mean a finite quotient with $B_n$ having cyclic image. Our main result is the following theorem.

Let $n \geq 5$ and $g \geq 1$. Suppose that $G$ is a finite nonabelian quotient of $B_n(\Sigma_g)$.

If $G$ is not braid-reduced then $|G| \geq n!$ with equality if and only if $G \iso S_n$.

If $G$ is braid-reduced then $G$ is 2-step nilpotent and $|G| \geq p^{2g+j}$, where $p$ is the smallest prime dividing $g+n-1$ and $j = 1$ or $2$ according to whether $p$ is odd or 2 respectively. Equality occurs if and only if either $G \iso \I(p^j,g)$ or $G \iso \II(p^j,g)$ (these two groups are nonisomorphic 2-step nilpotent $p$-groups defined in <ref>).

In particular the smallest non-nilpotent quotient of $B_n(\Sigma_g)$ is $S_n$.

Note that <ref> implies the following qualitative result.

* Fix $g \geq 1$. For all sufficiently large $n$, the smallest nonabelian quotients of $B_n(\Sigma_g)$ are 2-step nilpotent $p$-groups (in particular, the smallest nonabelian quotient is not $S_n$).

* Fix $n \geq 5$. For all sufficiently large $g$, the smallest nonabelian quotient of $B_n(\Sigma_g)$ is $S_n$. Also, there exists a (small) $g$ for which this not true.

[Smaller cases]

* If $n = 1,2,3,4$ and $g \geq 1$ (with the exception of $(n,g) = (1,1)$ where $B_n(\Sigma_g) = \pi_1(T^2) = \ZZ^2$ is abelian) then the symmetric group $S_3$ is the smallest nonabelian quotient of $B_n(\Sigma_g)$.

* If $g = 0$ then $B_n(\Sigma_g)$ is the spherical braid group $B_n(S^2)$ which is an intermediate quotient of the map $B_n \to S_n$ <cit.>. It follows from the result of Kolay <cit.> that the smallest quotient of $B_n(S^2)$ is $S_n$ for $n \geq 5$ and $S_3$ for $n = 3,4$. For $n = 1$ and $2$ we note that $B_n(S^2)$ is abelian.

From <ref> we obtain partial confirmation of a conjecture of Chen <cit.>:

Let $n \geq 5$ and $m \geq 3$, and let $g,h \geq 0$. If $n > m$ then there are no surjective homomorphisms
$$B_n(\Sigma_g) \to B_m(\Sigma_h).$$

Proof method.
<ref> follows from Kolay: By mapping a braid to its permutation on points, $S_n$ is a finite quotient of $B_n(\Sigma_g)$. If $B_n \to B_n(\Sigma_g) \to G$ has noncyclic image then $|G| \geq n!$ with the bound attained only by $G \iso S_n$.

The primary contribution of this paper is <ref>, which considers the braid-reduced quotients. We utilize a presentation of $B_n(\Sigma_g)$ (<ref>) due to Bellingeri <cit.> and assume that $B_n$ has cyclic image to reduce the relations and conclude that a braid-reduced quotient $G$ must be nilpotent. If we further assume that $G$ is a nonabelian braid-reduced quotient of minimum order then $G$ belongs to a class of nilpotent groups called JN2 groups (<ref>) which were classified by Newman in 1960 <cit.>. It then suffices to find the smallest JN2 groups which can be realized as a quotient of $B_n(\Sigma_g)$, a straightforward task given the concrete nature of Newman's classification.

<ref> provides a self-contained exposition of the classification of JN2 group. In <ref> we prove <ref>, as well as <ref>.

Acknowledgements. I am grateful to my advisor Benson Farb for continued support throughout this project and for detailed comments on many revisions of this paper, as well as for suggesting this problem in the first place. I thank Peter Huxford for useful discussions about braid groups and small $p$-groups, and for many helpful suggestions during the editing process. I also thank Dan Margalit for taking the time to read and comment on an earlier draft.

§ JUST 2-STEP NILPOTENT GROUPS

In this section we introduce and classify JN2 groups, a class of nilpotent groups which includes all minimal nonabelian braid-reduced quotients of $B_n(\Sigma_g)$.

A group $G$ is ıjust 2-step nilpotent (JN2) if $G$ is 2-step nilpotent (in particular, nonabelian) and every proper quotient of $G$ is abelian.

Finite JN2 groups admit a complete and explicit classification due to Newman <cit.>: Any finite JN2 group can be assigned a unique class $(p^j,m)$ where $p$ is a prime and $j$ and $m$ are positive integers; up to isomorphism, there are precisely two JN2 groups of a given class $(p^j,m)$. We will state and prove this classification theorem in <ref>, following the general ideas of <cit.>.

All JN2 groups will hereafter be assumed to be finite. The following proposition will allow us to define the class $(p^j,m)$ of a JN2 group.

A finite group $G$ is JN2 if and only if there exists a prime $p$ such that

* $G' \defeq [G,G]$ is cyclic of order $p$,

* the center $ZG$ is cyclic of order a power of $p$, and

* $G/ZG$ is elementary abelian of exponent $p$.

In particular, a JN2 group is a $p$-group.

Let $G$ be a finite JN2 group. For every nontrivial normal subgroup $N \trianglelefteq G$, we have that $G' \leq N$ since any proper quotient of $G$ is abelian. Since $G$ is 2-step nilpotent, $G' \leq ZG$. Consequently,

* $G'$ is abelian and admits no proper nontrivial subgroups so $G' \iso \ZZ/p\ZZ$ for some prime $p$.

* $ZG$ cannot be properly decomposed as a direct sum: Any nontrivial subgroup of $ZG$ contains $G'$ so no two nontrivial subgroups intersect trivially. Since $ZG$ is finite abelian, it must be cyclic of prime power order. The prime must be $p$ because $G' \leq ZG$.

* $G/ZG$ is abelian because $G' \leq ZG$. For $x,y \in G$, we have that $[x^p,y] = [x,y]^p$ by using the identity
$$[xz,y] = z[x,y]z^{-1}[z,y]$$
and noting that $[x,y]$ is central because $G' \leq ZG$. But $G'$ has order $p$, so in fact $[x^p,y] = 1$. Thus $x^p \in ZG$ for all $x \in G$, which is to say that $G / ZG$ has exponent $p$.

Suppose $G$ is a finite group satisfying (a), (b), and (c). Then $G' \neq \{1\}$ by (a) and $G' \leq ZG$ by (c) so $G$ is 2-step nilpotent.

If $N \trianglelefteq G$ is a normal subgroup with $G' \not \leq N$ then $N \cap G' = \{1\}$ by (a). Since $N$ is normal, $[N,G] \leq N \cap G' = \{1\}$ so $N \leq ZG$. But $G' \leq ZG$, and (a) and (b) imply that any nontrivial subgroup of $ZG$ intersects $G'$ nontrivially. Thus $N = \{1\}$. We conclude that every proper quotient of $G$ is abelian.

An immediate corollary of <ref> is that $V \defeq G/ZG$ has the structure of an $\FF_p$-vector space. Note that vector addition in $V$ is written multiplicatively and scalar multiplication of an element $x \mod ZG \in V$  by a scalar $r \in \FF_p$ is written as

\begin{gather*}
r \cdot (x \mod ZG) = x^r \mod ZG.
\end{gather*}

Fix a generator $z$ of $ZG$. This fixes a generator $z^{p^{j-1}}$ of $G'$ and hence an identification of $G'$ with $\FF_p$. Define a pairing

\begin{align*}
V \x V &\to G' = \FF_p \\
(x \mod ZG, y \mod ZG) &\mapsto [x,y]
\end{align*}

This pairing is a well-defined, bilinear, nondegenerate, alternating form which makes $V$ into a symplectic vector space. In particular, $\dim V$ is even.

Thus associated to each JN2 group $G$ is a class $(p^j, m)$ where $|ZG| = p^j$ and $\dim V = 2m$, so $G$ fits into the short exact sequence

\[
\begin{tikzcd}
1 \rar & \ZZ/p^j\ZZ \rar & G \rar & (\ZZ/p\ZZ)^{2m} \rar & 0.
\end{tikzcd}
\]

The symplectic structure on central factor groups $V = G/ZG$ is key to the classification theorem because symplectic automorphisms on central factor groups can  be used to construct isomorphisms between certain JN2 groups of the same class. The following lemma extracts from a JN2 group a normalized symplectic basis on its associated vector space $V$.

Let $G$ be JN2 of class $(p^j,m)$ where $p^j \neq 2$, with a fixed generator $z$ of $ZG$. Then there exists a symplectic basis $\sB = \{a_i \mod ZG, b_i \mod ZG\}_{i=1}^m$ of $V = G/ZG$ such that the representatives $a_i,b_i \in G$ satisfy either
[label=(*), font=]

* $a_i^p = b_i^p = 1$ for all $i$, or

* $a_1^p = b_1^p = z$ and $a_i^p = b_i^p = 1$ for $2 \leq i \leq m$.

We will say that $\sB$ is type I or II accordingly.

For the reader familiar with existing terminology from <cit.>, a “type I (II) basis” as named in our <ref> corresponds to a “canonic normal basis with zero (one) pairs of type II” in the vocabulary of Newman.

Note that $x^p \in ZG$ for all $x \in G$ because $G/ZG$ has exponent $p$. Let $(ZG)^p = \{ u^p : u \in ZG \}$ and identify $ZG / (ZG)^p$ with $\FF_p$ by mapping $z \mod (ZG)^p \mapsto 1$. Define a map

\begin{align*}
\nu: V &\to ZG / (ZG)^p = \FF_p \\
x \mod ZG &\mapsto x^p \mod (ZG)^p
\end{align*}

Viewing $V = G/ZG$ as a vector space written multiplicatively, $\nu$ commutes with scalar multiplication and

\begin{gather*}
\nu((x \mod ZG) (y \mod ZG)) = (xy)^p \mod (ZG)^p = [y,x]^{\frac{p(p-1)}{2}} x^p y^p \mod (ZG)^p
\end{gather*}

for $x,y \in G$ so $\nu$ is a linear functional as long as $[y,x]^{\frac{p(p-1)}{2}} = 1 \mod (ZG)^p$. This holds if $p^j \neq 2$: If $p$ is odd then $p \mid \frac{p(p-1)}{2}$ so $[y,x]^\frac{p(p-1)}{2} = 1$ because $G'$ has order $p$. If $j \geq 2$ then $G' \nleq ZG$ so $G' \leq (ZG)^p$.

If $\nu$ is the trivial linear functional on $V$, take $\sB$ to be any symplectic basis of $V$. Otherwise, there exists a symplectic basis $\sB$ of $V$ such that $\nu$ written with respect to $\sB$ is the row vector

\begin{gather*}
\nu = \m{1 & 1 & 0 & \cdots & 0}
\end{gather*}

because symplectic automorphisms act transitively on nontrivial vectors.

In other words, for each basis vector $x_j \mod ZG \in \sB$,

$$x_j^p \mod (ZG)^p = \nu(x_j \mod ZG) = z^{\nu_j} \mod (ZG)^p$$

so there exists $u_j \in ZG$ such that $x_j^p = z^{\nu_j} u_j^p$. Then $x_ju_j^{-1} \equiv x_j \mod ZG$ and $(x_ju_j^{-1})^p = z^{\nu_j}$. Thus $x_ju_j^{-1} \in G$ are representatives of the basis $\sB$ satisfying (I) if $\nu$ is trivial and (II) otherwise.

We will now construct two standard non-isomorphic JN2 groups for each given class $(p^j,m)$. The proof of the classification theorem will exhibit an isomorphism from any arbitrary JN2 group to a standard one. The primary method of constructing larger JN2 groups from smaller ones is taking a central product.

Let $G$ and $H$ be groups for which there exists an isomorphism $\varphi: ZG \to ZH$. Define the ıcentral product of $G$ and $H$ (with respect to $\varphi$) to be

\begin{gather*}
G \odot H = (G \x H) / N
\end{gather*}

where $N = \ang{(g,\varphi(g)^{-1}): g \in ZG}$, namely identifying $ZG \x 1$ with $1 \x ZH$ by the isomorphism $\varphi$. By $G^{\odot n}$ we mean the central product of $n$ copies of $G$ with the identity isomorphism on $ZG$.

Note that if $G,H$ are JN2 of class $(p^j,m_1)$ and $(p^j,m_2)$ then $G \odot H$ is JN2 of class $(p^j, m_1 + m_2)$ by <ref> since

* $(G \odot H)' = G' \x H' / N \iso G' \iso H' \iso \ZZ/p\ZZ$,

* $Z(G \odot H) \iso ZG \iso ZH$, and

* $(G\odot H)/Z(G \odot H) \iso (G/ZG) \x (H / ZG)$.

[Standard JN2 groups]
Define the groups

\begin{align*}
M(p^j) &= \ang{z, a, b: [z,a] = [z,b] = 1; [a,b] = z^{p^{j-1}}; z^{p^j} = a^p = b^p = 1} \\
N(p^j) &= \ang{z, a, b: [z,a] = [z,b] = 1; [a,b] = z^{p^{j-1}}; z^{p^j} = 1; a^p = b^p = z} \\
\I(p^j,m) &= M(p^j)^{\odot m} \\
\II(p^j,m) &= N(p^j) \odot M(p^j)^{\odot (m-1)}
\end{align*}

Observe the following:

* $M(p^j)$ and $N(p^j)$ are JN2 (by <ref>) of class $(p^j,1)$ with each center generated by $z$ and $\{a,b\}$ as a symplectic basis of $V$.

* $\I(p^j,m)$ and $\II(p^j,m)$ are JN2 of class $(p^j,m)$ by the remarks following <ref>.

* $\I(p^j,m)$ and $\II(p^j,m)$ are not isomorphic when $p^j \neq 2$: The group $N(p^j)$ has an element of order $p^{j+1}$ (for example, $a$ or $b$) and therefore so does $\II(p^j,m)$. On the contrary, the group $M(p^j)$, and consequently also $\I(p^j,m) = M(p^j)^{\odot m}$, has exponent at most $p^j$: The linear functional $\nu$ (as in the proof of <ref>) is trivial on the symplectic basis $\{a,b\}$ so $M(p^j)^p \leq (ZG)^p$, hence $M(p^j)^{p^j} \leq (ZG)^{p^j} = 1$.

Note: If $p^j = 2$, then $\I(p^j,m)$ and $\II(p^j,m)$ are still non-isomorphic: $M(2)$ is the dihedral group $D_8$ and $N(2)$ is the quaternion group $Q_8$, which contain two and six elements of order 4 respectively and both have centers of order 2. In particular no elements of order 4 are central. The larger groups $\I(p^j,m)$ and $\II(p^j,m)$ can then be distinguished by counting the number of elements of order 4 because only central elements are identified in the central product. We will not require this case.

We are now ready to state and prove the classification theorem of JN2 groups.

Let $G$ be JN2 of class $(p^j,m)$. Suppose that $p^j \neq 2$. Then $G$ is isomorphic to either $\I(p^j,m)$ or $\II(p^j,m)$.

Let $z$ be a generator of $ZG$ and let $\sB$ be the symplectic basis given by <ref>. In the notation of <ref>, let $H_i = \ang{z,a_i,b_i}$. If $\sB$ is type I then $H_i = M(p^j)$ for all $i$. If $\sB$ is type II then $H_1 = N(p^j)$ and $H_i = M(p^j)$ for $i \geq 2$.

The subgroups $H_i$ commute pairwise, together generate $G$, and intersect precisely in their centres $\ang{z}$, so $G \iso \bigodot_{i=1}^m H_i$. Hence $G$ is isomorphic to $\I(p^j,m)$ or $\II(p^j,m)$ according to the type of the basis $\sB$.

* (Generalizations)
For brevity, we have excluded the case of $p^j = 2$ and specialized to finite groups. With additional work, the $p^j = 2$ case and some infinite JN2 groups (those with a countable symplectic basis) also admit a classification as central products of elementary JN2 groups, see <cit.>.

* (Special cases)
Note that $M(p)$ and $N(p)$ are the only two groups of order $p^3$. The group $M(p) = \I(p,1)$ is isomorphic to the Heisenberg group over $\FF_p$. A generalization of the finite Heisenberg groups are the extraspecial groups, which are defined to be $p$-groups $G$ with $ZG$ order $p$ and $G/ZG$ nontrivial elementary abelian. In particular, extraspecial groups are JN2 and it follows from <ref> that there are precisely two distinct extraspecial groups of order $p^{1+2m}$ for each choice of a prime $p$ and positive integer $m$, and that this exhausts all extraspecial groups.

§ MINIMAL NONABELIAN QUOTIENTS OF $B_N(\SIGMA_G)$

In this section we provide the proof of <ref>.

The strategy of the proof will be to utilize an explicit presentation of the surface braid groups (<ref>) to characterize braid-reduced quotients by the relations that they must satisfy (<ref>). We will then show that many JN2 groups are realized as nonabelian braid-reduced quotients of $B_n(\Sigma_g)$ (<ref>) and finally prove that all nonabelian braid-reduced quotients of minimum order belong to the list of JN2 groups in <ref>.

The following presentation of $B_n(\Sigma_g)$ is due to Bellingeri <cit.>.

For $g \geq 1$ and $n \geq 2$, the surface braid group $B_n(\Sigma_g)$ admits the presentation:

* generators: $\sigma_1,\dots,\sigma_{n-1},a_1,\dots,a_g,b_1,\dots,b_g$.

* relations:

braid relations:

[σ_i, σ_j] = 1       (1 ≤i,j ≤n-1  and  |i-j| ≥2)

σ_iσ_i+1σ_i = σ_i+1σ_iσ_i+1       (1 ≤i ≤n-2)

mixed relations:
(R1)         [a_r, σ_i] = [b_r, σ_i] = 1
(1 ≤r ≤g  and  i ≠1)

[a_r, σ_1^-1a_rσ_1^-1] = [b_r, σ_1^-1b_rσ_1^-1] = 1
(1 ≤r ≤g)

[a_s,σ_1a_rσ_1^-1] = [b_s,σ_1b_rσ_1^-1] = 1
(1 ≤s < r ≤g)

[b_s,σ_1a_rσ_1^-1] = [a_s, σ_1b_rσ_1^-1] = 1
(1 ≤s < r ≤g)

[a_r, σ_1^-1b_rσ_1^-1] = σ_1^2
(1 ≤r ≤g)

[a_1,b_1^-1] ⋯[a_g,b_g^-1] = σ_1σ_2⋯σ_n-1^2 ⋯σ_2σ_1

The embeddings $B_n \hookrightarrow B_n(\Sigma_g)$ identify the Artin braid generators with the Bellingeri generators $\sigma_i$. The remaining generators $a_r,b_r$ can be understood loosely to be the standard generators of $\pi_1(\Sigma_g)$.

More precisely, let $\{p_1,\dots,p_n\} \in \UConf_n(\Sigma_g)$ denote the basepoint of $B_n(\Sigma_g)$ and let $D \subset \Sigma_g$ be an open disk with $p_1 \in \del D$, with $p_2,\dots,p_n$ in the interior of $D$. There is an inclusion

$$ \pi_1(\Sigma_g - D, p_1) \hookrightarrow B_n(\Sigma_g) $$

which takes a loop $\gamma$ in $\Sigma_g - D$ to the braid on $\Sigma_g$ with first strand $\gamma$ and all other strands trivial. The group $\pi_1(\Sigma_g - D, p_1)$ is free on $2g$ generators and surjects onto $\pi_1(\Sigma_g, p_1)$ which has a standard presentation. The surface braid group generators $a_r,b_r \in B_n(\Sigma_g)$ can then be understood as a choice of a free generating set of $\pi_1(\Sigma_g - D, p_1)$ which lifts the standard generating set of $\pi_1(\Sigma_g, p_1)$. It should be emphasized that the lifts are not canonical and that the presentation depends on the choices; the curious reader may refer to <cit.> for illustrations of the loops which produce this particular presentation.

Let $n \geq 3$ and $g \geq 1$. A finite group $G$ is a braid-reduced quotient of $B_n(\Sigma_g)$ if and only if $G$ admits a generating set $\{\sigma,a_1,b_1,\dots,a_g,b_g\}$ satisfying the relations

R1         [a_r, σ] = [b_r, σ] = 1
(1 ≤r ≤g)

[a_s,a_r] = [b_s,b_r] = [b_s,a_r] = [a_s, b_r] = 1         (1 ≤s < r ≤g)

[a_r, b_r] = σ^2
(1 ≤r ≤g)

σ^2(g+n-1) = 1

A finite quotient of $B_n(\Sigma_g)$ is presented by <ref> with additional relations. The condition that $B_n$ has cyclic image in a quotient is equivalent to adding the relations
$$\sigma_i = \sigma_1,\qquad 1\leq i \leq n.$$
If we add these relations and write $\sigma = \sigma_1$, the relation (R2) is made redundant and (R1), (R3), and (R4) respectively reduce to the relations R1, R3, and R4 as in the statement of the lemma. The final relation (TR) reduces to

\begin{gather*}
[a_1,b_1^{-1}] \cdots [a_g,b_g^{-1}] = \sigma^{2(n-1)}
\end{gather*}

which is equivalent to TR because from R4 we can write $a_r = b_r^{-1} \sigma^{-2}a_r b_r$ so that

\begin{gather*}
[a_r, b_r^{-1}]
= a_r b_r^{-1} a_r^{-1} b_r
\overset{\text{\Grel{R4}}}{=}
(b_r^{-1} \sigma^{-2} a_r b_r) b_r^{-1} a_r^{-1} b_r
= b_r^{-1} \sigma^{-2} b_r
\overset{\text{\Grel{R1}}}{=}
\sigma^{-2}. \qedhere
\end{gather*}

The following lemma proves that many JN2 groups are braid-reduced quotients.

Let $n \geq 3$ and $g \geq 1$. Let $p$ be a prime dividing $g+n-1$.

* If $p = 2$ then $\I(2^2,g)$ and $\II(2^j,g)$ for all $j \geq 2$ are nonabelian braid-reduced quotients of $B_n(\Sigma_g)$.

* If $p$ is odd then $\I(p,g)$ and $\II(p^j,g)$ for all $j \geq 1$ are nonabelian braid-reduced quotients of $B_n(\Sigma_g)$.

Let $p$ be a prime dividing $g+n-1$. By <ref> we need to exhibit a generating set $\{\sigma,a_r,b_r\}$ of each group satisfying relations R1, R3, R4, and TR.

In any of the JN2 groups in the statement of the theorem, fix a generator $z$ of the center and choose $a_1,b_1,\dots,a_g,b_g$ to be the representatives of a symplectic basis of $V$ given by <ref>. By <ref> this basis will be type I for $\I(2^2,g)$ and $\I(p,g)$, and type II for $\II(2^j,g)$ and $\II(p^j,g)$. Note that with the given symplectic form, the condition that a basis is symplectic is simply that all basis elements commute except symplectic pairs $[a_r,b_r] = z^{p^{j-1}}$. In particular, R3 is satisfied.

We will now choose $\sigma$ for each group and verify that $\{\sigma,a_r,b_r\}$ generate the group and satisfy R4.

* $\I(2^2,g)$ is generated by $\sigma = z$ and the $a_r,b_r$. These generators satisfy R4 because $[a_r,b_r] = z^2 = \sigma^2$.

* $\II(2^j,g)$, for a given $j \geq 2$, is generated by the $a_r,b_r$ alone because $a_1^p = z$. If we choose $\sigma = z^{2^{j-2}}$ then R4 is satisfied because $[a_r, b_r] = z^{2^{j-1}} = \sigma^2$.

* $\I(p,g)$ for odd prime $p$ is generated by $\sigma = z^{(p^j+1)/2}$ and the $a_r,b_r$. Then R4 is satisfied because $[a_r,b_r] = z = \sigma^2$.

* $\II(p^j,g)$, for given odd prime $p$ and $j \geq 1$, is generated by $a_r,b_r$ alone because $a_1^p = z$. Then set $\sigma = z^{(p^j+p^{j-1})/2}$ so that R4 is satisfied because $[a_r,b_r] = z^{p^{j-1}} = \sigma^2$.

In all cases $\sigma$ was chosen to be central, hence R1 is satisfied.

It remains to check that TR holds, namely that $|\sigma|$ divides $2 (g+n-1)$. Recall that we are assuming that $p \mid (g+n-1)$. In cases 1 and 2, we have $p = 2$ and $|\sigma| = 4 = 2p \mid 2(g+n-1)$. In cases 3 and 4, we have $|\sigma| = p \mid (g+n-1)$.

We are now prepared to prove <ref>.

Let $G$ be a nonabelian braid-reduced quotient and let $\{\sigma,a_1,b_1,\dots,a_g,b_g\}$ denote the generating set of $G$ as given by <ref>. By R1 and R3, all pairs of these generators commute except for pairs $a_r,b_r$ so $G' = \ang{\sigma^2}$ by R4. Then $G'$ is central and nontrivial, which is to say that $G$ is 2-step nilpotent.

Assume now that $G$ is of minimum order. Then $G$ has no proper nonabelian quotients and thus is JN2 of some class $(p^j,m)$.

We make three claims:

* $p^j \neq 2$,

* $m = g$, and

* $p \mid (g+n-1)$.

These claims will complete the proof: Since $|G| = p^{2m+j}$, it follows from claims 1 and 2 that $|G| \geq p^{2g+1}$ if $p$ is odd and $|G| \geq 2^{2g+2}$ if $p = 2$. Claims 1 and 3 along with the minimality of $G$ together imply that $G$ is one of (in particular, the smallest of) the quotients constructed in <ref>. Explicitly: If $g+n-1$ is even then $p^j = 2^2$. Otherwise $p^j = p$ where $p$ is the smallest prime dividing $g+n-1$. Finally, $G$ must be isomorphic to either $\I(p^j,g)$ or $\II(p^j,g)$ by <ref>.

Proof of claims:
Let $d = |\sigma|$. By R1, $\sigma$ is central so $d \mid p^j$. But $p = |G'| = |\sigma^2|$ so $d \mid 2p$. Thus either $p$ is odd and $p = d$, or $p = 2$ and $d = 4$.

* If $p$ is odd then $p^j \neq 2$. If $p = 2$ then $p^j \geq d = 4$ so $p^j \neq 2$.

* We will show that $\dim V = 2g$ by proving that

$$\sB = \{a_r \mod ZG, b_r \mod ZG\}_{r=1}^g$$

is a basis of $V$. Every element $x \in G$ can be written uniquely in the form

$$x = \sigma^k a_1^{i_1} \cdots a_g^{i_g} b_1^{j_1} \cdots b_g^{j_g}$$

using commuting relations R1, R3, R4 so $\sB$ is a generating set. To prove that $\sB$ is linearly independent, let

$$y = a_1^{i_1} \cdots a_g^{i_g} b_1^{j_1} \cdots b_g^{j_g} \in G$$

and suppose that $y = 0 \mod ZG$, which is to suppose that an arbitrary linear combination of elements of $\sB$ is trivial in $V$. Then $y$ is central so

$$[y,b_1] = [a_1^{i_1},b_1] = \sigma^{-2i_1} = 1$$

which implies that $d \mid 2 i_1$ and thus $i_1 = 0 \mod p$: If $p$ is odd then $d = p$ so $p \mid i_1$. If $p = 2$ then $d = 4 \mid 2 i_1$ so $p = 2 \mid i_1$.

Similarly $i_r = j_r = 0 \mod p$ for all $r$, which is to say that all coefficients of the linear combination are trivial over the base field $\FF_p$. This proves the linear independence of $\sB$.

* The relation TR imposes the relation $d \mid 2(g+n-1)$. Either $d = p$ is odd or $d = 4$ and $p = 2$; in both cases TR implies that $p \mid (g+n-1)$.

Let $n \geq 5$ and $m \geq 3$, and let $g,h \geq 0$. If there is a surjection $B_n(\Sigma_g) \to B_m(\Sigma_h)$ then the composition $B_n(\Sigma_g) \to B_m(\Sigma_h) \to S_m$ is also surjective. Since $S_m$ is not nilpotent when $m \geq 3$, we must have $m \geq n$.

In his paper <cit.>, Bellingeri also gives a presentation of the braid group of a genus $g$ surface with $m$ punctures (equivalently for the purposes of braid groups, $m$ boundary components). The above methods can be used nearly verbatim to prove that the smallest nonabelian quotient of $B_n(\Sigma_{g,m})$ is the smaller of $S_n$ or $\I(2^2,g)$ and $\II(2^2,g)$.